I can give a partial answer here.
TvaritA is so called because she gives results quickly (from the word tvarita meaning quick).
I know her mode of worship which is Tantric in nature and the relevant verses are quoted in the Brihat TantrasArah. The text however does not specify which of the Tantras it is quoting from.
Atha tvaritA mantrah ||
athAbhidhAsye tvaritAm tvaritam phaladAyinim |
varma-bijam riddhi-rishasvarAsvitA ||
Samvarto bhagavAn maya-phatanto
MAyA bhuvanshi varma hum riddhi kha-karah
ishwarah e-kArah kurmashcha-kArah tadantash-chakArah bhaga
ekarastada-yuktah kshah stri swarupam dirgha-tanucchadam hum
samvartakah ksha-kArah bhaga e-kArastada-yuktah punarbhuvaneswari phat
These verses at the start is saying that quick results follow if one chants the TvaritA mantra. And then the verses are giving the mantra in a coded form.
tvaritAm tvaritam phaladAyinim -- Goddess TvaritA gives quick results. So, this gives you an idea why she is so called.
The decoded mantra is -- Om hrim hum khechache kshastri hum kshe hrim phat
It's a 12-letter mantra whose Rishi (seer) is stated to be Arjuna (from MahAbhArata).
AsyArjunarishi-virAta chandah tvaritadevatA purushArtha-chatushtaya
siddhyarthe viniyogah |
For the mantra, Arjuna is the Rishi, VirAt is the Chandah and TvaritA
is the DevatA. For accomplishing the four goals of life, the chanting
is being done.
Then the method her worship is given. DhyAna sloka, NyAsa mantras etc are given in the text. The Purascharana of the mantra is accomplished by one lakh chants.
The Puja is to be done using the deity's Yantra which looks like:
In the Yantra, inside all the petals the Bija Hum is to be inscribed whereas in the circle the Bija Hrim. And, inside the circle, clockwise, all the 50 letters of Sanskrit are also to be inscribed.
The Devi's description is detailed in her DhyAna sloka and which is as follows:
ShyAmAm vahir-kalApa-shekhara-yutAmA-vaddha-parnAmshukAm |
Gunja-hAralasat-payodharanatAmashtAhipAn vibhratim ||
TArankAngad-mekhalA-guna-ranan-manjiratAm prApitAn |
varadAbhayodyatakarAm devim trinetrAm bhaje ||
mAnaseyih sampujya shankha-sthApanam kuryAt ....
The Devi is having dark complexion. She has a crown in the head which
is having peacock's tail. Her garments are made of leaves. She is
wearing necklace that is made of Gunja flowers (rosary pea). The hem
of the cloth which is made of eight snakes is hanging down to the
weight of her breasts. She is decked with armlets, waist-chain and
anklets. She is having three eyes and has the appearance of a KirAta. On
one hand she shows the boon-giving gesture and on the other the
Upon meditating thus one has to do the mental worship and thereafter
establish the conch etc ...
And, as shown in this answer, Devi TvaritA belongs to the KAli Kula and not to the Sri Kula.