Are those horses related to Atharva Veda Book 19 Hymn 53?
1 Prolific, thousand-eyed, and undecaying, a horse with seven reins Time bears us onward.
2 This Time hath seven rolling wheels and seven naves immorality is the chariot's axle.
Is God Surya or those horses in any way related to Kala (time)?
Who is the God sitting in front of Surya, controlling those horses with stick?
What is that stick called?
Which other God or any Character from scriptures ride on seven horse or seven horse chariot?
There can be 2 explanations that are coming to my mind, at present.
1. Post Vedic/Puranic/General
a) 7 horses may indicate 7 colours of the Nature. Light that emanates from the Sun or Surya contains 7 colours.
b) The person that is sitting before Surya is Aruna, the elder brother of Garuda.
Their mother Vinata was out of jealousy broke one of 2 eggs and found in it an embryo with the upper part developed but the lower one undeveloped.
At this, the child in the egg became angry and cursed his mother, saying.
'Since thou hast prematurely broken this egg, thou shall serve as a slave. Shouldst thou wait five hundred years and not destroy, or render the other egg half-developed, by breaking it through impatience, then the illustrious child within it will deliver thee from slavery! And if thou wouldst have the child strong, thou must take tender care of the egg for all this time!'
Thus cursing his mother, the child rose to the sky. O Brahmana, even he is the charioteer of Surya, always seen in the hour of morning!
2. Vedic approach (from Rig Veda)
The Vedic horse (ashva) is a symbol of the life energy (prana) and indicates perfect control over the breath (Pranayama) as well as dazzling speed.
In Rig Veda, yoking of chariot was mentioned, indicating Yoga.
The following verse from Atharva veda was mentioned by the OP in the question.
This Time hath seven rolling wheels and seven naves immorality is the chariot's axle.
Here, seven naves indicate 7 chakras in Yogic way and immortality indicates, merging with BRAHMAN and attaining BLISS, but not in physical sense of immortality.
So starting the journey from Muladhara and passing through remaining chakras, and ends in BLISS.
The Sanskrit word अश्व is used here. Yaska in nirukta 2:27 explain it as one who move in it's way. It can be anything. Many interpret it as Ray's.
Rigved 1:105:9:- Where those seven rays of sun are shining, thence my home and family extend.This Tṛta Āptya knoweth well, and speaketh out for brotherhood. Mark this my woe, ye Earth and Heaven.
The verse cleared that Hindus knew about sun's seven Ray's.
In some other scriptures, Ray's are described as Vedic meters or chandas.
SB 5.21.15: My dear King, the carriage of the sun-god's chariot is estimated to be 3,600,000 yojanas [28,800,000 miles] long and one-fourth as wide [900,000 yojanas, or 7,200,000 miles]. The chariot's horses, which represents Gāyatrī and other Vedic meters, are harnessed by Aruṇadeva to a yoke that is also 900,000 yojanas wide. This chariot continuously carries the sun-god.
While describing the chariots of all the nine planets, Lord Vishnu told Rudra–’ The chariot of Surya deva(sun) has an unbelievable expanse stretching up to nine thousand yojans. The axle of the chariot is one crore and fifty seven lakh yojans long and wheels are fixed at both its ends. The wheels have six circumferences symbolizing the six main seasons and five spokes symbolizing the five different units of time measurement in them. The chariot of Surya(sun) is pulled by seven horses. These horses symbolize the seven ‘chhands’ (stanzas used in poetry)- gayatri, vrihati, ushnik, jagati, trishtup, anushtup and pankti. Garuda Purana,
The two halves of the yoke are of the same length respectively as the two axles (the longer and the shorter). The short axle, with the short yoke, are supported by the pole-star: the end of the longer axle, to which the wheel of the car is attached, moves on the Mānasa mountain5. The seven horses of the sun's car are the metres of the Vedas, Gāyatrī, Vrihatī, Uṣṇih, Jayatī, Tṛṣṭubh, Anuṣṭubh, and Pankti.
46-48. The chariot is yoked with only seven horses. They are the Veḍic Metres that have assumed the form of horses. They are stationed towards the place where the wheel is fitted. They assume the forms as they wish. They are green, imperishable; they (? are accompanied) by tawny-coloured masters expounding the Brahman. In the course of a year those horses traverse eighty-three hundred circles.
Atharvaved 13:2:39 says that sun is reference of time.
It is a fact that sun decides the day, night, months, seasons and years.
He is Aruna, Elder brother of garuna and son of sage Kashyap and Vinata. He became charioter of sun because of their father's boon.
SB 5.21.16: Although Aruṇadeva sits in front of the sun-god and is engaged in driving the chariot and controlling the horses, he looks backward toward the sun-god.
And she gave birth to two sons, Aruna and Garuda. And Aruna, of undeveloped body, became the fore-runner of the Sun. And Garuda was vested with the lordship over the birds. O you of Bhrigu’s race, hearken now to the mighty achievement of Garuda.'"
Scriptures have know reference about stick. Some pictures show that stick or some not. Scriptures define about the dimensions of chariot, spokes of wheel and even about the axle of wheel, but no scripture talks about that stick.
The chariot of the sun is nine thousand leagues in length, and the pole is of twice that longitude1; the axle is fifteen millions and seven hundred thousand leagues long2; on which is fixed a wheel with three naves, five spokes, and six peripheries, consisting of the ever-during year; the whole constituting the circle or wheel of time4. The chariot has another axle, which is forty-five thousand five hundred leagues long5.
The two halves of the yoke are of the same length respectively as the two axles (the longer and the shorter). The short axle, with the short yoke, are supported by the pole-star: the end of the longer axle, to which the wheel of the car is attached, moves on the Mānasa mountain5. The seven horses of the sun's car are the metres of the Vedas, Gāyatrī, Vrihatī, Uṣṇih, Jayatī, Tṛṣṭubh, Anuṣṭubh, and Pankti.
Chariot of seven horses is associated with Lord Surya but other planets like Chandra rides on the chariots of different no.s of horses like chariot of Chandra has 10 horses.
The car of the moon has three wheels and the horses are white as Kunda flowers. It runs, drawn by ten horses, on the right and left. The car of the son of the moon (Budha) is made of [the essence of] air and fire. It is drawn by eight yellow-coloured steeds fleet as the wind. The great chariot of Shukra has a Varutha,2 Anukarsha,4 is drawn by horses born of earth and adorned with flags. The huge chariot of Bhumi’s (earth) son (Mars) is of the colour of molten gold and is drawn by eight steeds, of the colour of the filaments of a lotus and born of fire. Jupiter resides for one year at every sign of a Zodiac sitting on his golden car drawn by eight yellowish-white horses. Riding a car drawn by horses of variegated colour and born of ether Saturn moves slowly on; Svarbhanu5 has eight horses of the colour of the earth and his car is grey-hued. O lord of goblins, yoked to his car they carry him, day and night. Rahu’s car has eight horses, fleet as the wind and smoky-coloured and which have their tongues coated with saliva. On it he roams over the earth consisting of islands, rivers and mountains.
The other answers here have missed a really simple point, because of their lack of awareness of Jyotisha. How can you even comment anything about Surya Bhagavaan or Aadityas without knowing Jyotisha? It's ridiculous.
But anyways, the answer is really simple.
Are those horses related to Atharva Veda Book 19 Hymn 53?
Yes.
Is God Surya or those horses in any way related to Kala (time)?
Yes The sun rises and sets. In terms of the natural Zodiac, or Kaala Purusha's Jaataka, the sun rises in the Mesha (Aries) Raashi and sets in the Tulaa (Libra) Raashi. The sun rises in a counter zodiac order (aka clockwise). If you notice, these are 7 in number. Each raashi takes roughly 2 hours to pass. Surya bhagavaan takes 12 hours, roughly to rise and set. Since he rises in the middle of the Aries, and sets in the middle of Libra, this works out as 6 raashi lengths total (12 hours), despite there being 7 horses aka raashis.
The first horse represents 6-7am, the second 7-9, the third 9-11, the fourth 11-1, the fifth 1-3, the sixth 3-5 and the seventh 5-6 at which point, Surya bhagvaan sets.
Those horses are the 7 positions of the 7 raashis. NOT the seven raashis themselves. As, the sun remains in one raashi for one whole month, it's just that the raashi, along with the sun moves through those 7 positions.
Let me give you another evidence for this simple fact. The image you've provided, the second horse from the right is deep brown, and the 5th from right is black, correct?
Are you able to notice that from the right side, the order fits perfectly? The second (11th) is deep brown aka color of Aquarious (Kumbha) and the 5th (8th) is black aka color of Scorpio (Aali/Vrischika).
Now of course, this artist here in the picture: 
has messed it up due to his lack of knowledge... hinduism has a huge issue of intellectual incompetence, and it's at show here. The 3rd horse should've been deep brown, not the second and similarly, the 6th should've been black, not 5th.The the colors of the other Raashis aren't provided either. But anyways, that's not your question.
Which other God or any Character from scriptures ride on seven horse or seven horse chariot?
None but Surya, to my knowledge.
Your other questions have already be answered, so I will leave it.
Also, as far as the 7 horses being 7 primary colors, that's asinine. Since the chariot is mentioned as golden in many places alongside the 7 horses, and Aruna means red itself. So would that make it 9 colors? No, there's no relation to colors here, other than the colors of the raashis itself.
