This month being Karthika 2 earthen lamps will be lit and kept in the Pirai Maadam [in old types houses there will be a construction like this for keeping lit earthern lamps] during sunset. Nowadays the lamps are kept in the entrance of the front door. Similarly in Margasisha month 2 earthen lamps will be lit and kept in the pirai Maadam by 5 AM. Any significance for this thru puranas or scriptures.
Importance of Karthika Deepa Dana(Importance of lighting lamps in Karthika month):
Karthika Dana Mahima(Importance of Charity during Karthika month):
Kartika Dana Mahima: In observing a Vrata in the Month of not eating in a Kamsya or bronze vessel, the person following th e Vrata should give a daana of five broze vessels full of ghee at the end of the Vrata; if honey i s left out then ghee, sugar and kseeraana should b e given away;in Tila ‘thyaga’ tila daana; in mou na bhojana vrata or eating in silence then ghanta (Bell) daana; in case of Phala varjana then Phala rasa daana; for dhanya varjana go daana and so on
But in that month, among all Danas, Deepa Dana is considered the best.
But nothing like Deepa daana : Ekatassarvva daan aani deepa daanam tathai kathah, Kartika deepa daa nasya kalaam naarhanti shodashim/ ( All types of Daanaas would equate one ‘Deepa Daana’; in Kartik a Month deepa daanaa might not be like sixteenth o f any daana!)
Source-Essence of Dharma Sindhu(The essence of the ocean of Dharma)
Also,in particular, Deepa Dana from Karthika Krishna Trayodashi at the Pradosha time (evening) till the end of the month are meant for Yama,the God of Death.
From Karthika Mahatmya Chapter of the Skanda Purana:
Lord Brahma affirmed that the fortnight from Kartika Krishna Trayodasi to the end of the month ie from Deepavali on the Amavasya of Aswayuja should be observed as Festival Days. The ‘Pradosha’ time of that Trayodasi is the most significant time when Lord Yama is worshipped with ‘Deepas’ at the main door of the residence and Naivedya to ensure that there would never be an ‘Apamrutyu’ or Akaala Mrithyu (premature and sudden death) in the family. Yama Raja would be pleased with the recitation of the Mantra after the worship: Mrityuna pasha dandaabhyaam Kaalena cha maya saha, Trayodashyam Deepa daanaath Suryajah Preeyathamit
Significance of lighting lamps at early morning in the Margasirsha month:
Not 100% sure,but these lamps are possibly dedicated to Lord Vishnu as we can guess from this chapter of Skanda Purana:
While addressing a group of Munis, Sage Sutha narrated that Lord Brahma asked Bhagavan Narayana about the significance of Margasirsha Month and the reply was that in His earlier incarnation as Krishna, He advised thousands of Gopikas and Gopas to observe austerities during that month by performing ‘Snaans’, ‘Pujas’, fastings and total dedication to Him to attain Salvation.
Thus Gopikas and Gopas adopted a daily routine of waking up early; taking bath with Tulasi roots of earth on head by reciting the Mula Mantra.... After the former bath and wearing clean clothes, the routine continues with applying ‘Vibhudi’ (ash of homa or of cow dung), perform ‘Sandhya Vandan’/ Gayatri Japa followed by Vishnu Puja with ‘Ghantanaad’ (ringing bell), Chandana, Pushpa, Tulasidal, dhoop, deepa and other ‘Shodasopacharas’, ‘Stutipaath’ (Extolling hymns) of Vishnu Sahasranaam
So,you can consider this as a partial answer.