Since they don't allow tripods, I would presume not. Although still photography with a DSLR is allowed, Archeological Survey of India sites don't allow tripods as you need to get a written permission from ASI to "film".

Better to avoid going to public places such as this, during this time.

visiting timings are from 7:00 am to 5:00 pm the best time to visit is during the Indian classical music festival which usually happens during the winter months of November or December.

Yes. Close to the monument

Its a bit cold but just a jacket will do, the problem is the rains

Not 100%

Yes same cost on all sundays

Taj Mahal, Mathura and Vrindavan can be seen in one day. Best option is to hire a cab. First visit Taj, then Agra Fort and while returning visit Mathura and Vrindvan. Prem Mandir shall be visited only by night. Lighting is too good. For Delhi half day for Akshar Dham temple. Reach the temple around 3 pm, if possible have a late lunch there. Visit the temple then must see the light and sound show. You will remember it for your whole life. It will start in the evening. Other part of Delhi can be seen in a day. Lal quila, Raj Ghat, Jama Masjid, India Gate, Humayun Tomb, Qutub Minar, Lotus Temple. Don't shop at Palika bazar, instead try Sarojini market or Lajpat market. Besides there are many malls. Try food at Andhra Bhavan, go for thali. In Delhi you can hire cab for a package OR you can easily use auto to move from one place to another. Online cab service is good in Delhi.

Nothing that I would eat at

500 Indian rupees.