Okay, let me answer you as best I can:
I never visited Udapiur, instead I went to Jaipur, Jaislamer, Jodphur, Bikaner (the famous rat temple) and Pushkar (for the camel fair). When I was in India, the water in Udapiur was very low and it detracted from the beauty of the place (according to travellers who had just been there).
The triangle is a good idea as it allows you to get a great snapshot of that part of the world. I'll borrow something I wrote on another thread here: In Agra, apart from the Taj, make sure you see Fatehpur Sikri and the rarely visited Itmad-ud-daulah's tomb - known as the mini-Taj - and I found the intricate carvings and adornments of that building to be better than its famous larger 'sibling'. Jaipur's stops should include the City Palace and my favourite, the Amer Fort.
From Delhi, there are plenty of trains that go overnight to Mumbai - some are much faster than others. Either go 2AC or 3AC class. With four of you, it will be easier to go for a 2AC as you will take up a whole 'cabin', assuming you can get all tickets for the same area. Rajdhani Express is the fastest.
Vaccinations are numerous - you need the usual, Hep A, Hep B, Tetanus, Typhoid treatments, and there is a oral medication (Dukoral) you can take that can protect against Cholera, with the other benefits is that it provides partial protection against Delhi Belly. Malaria medication is required here, so you'll need to start of course of tablets (I prefer Doxycycline) for that, and you are supposed to get Rabies shots before you go, but not many travellers (me included) bother with this.
As for what you wear, I am particularly conservative in this regard, I always wear long pants and long sleeves anywhere I go - even if the temperature gets into the 40-50 degree Celsius range. I'd be a bit cautious about exposing too much skin, only because it is not regarded as the norm according to local standards. Having said this, shorter sleeve tops are a better bet than shorter skirts.
New Delhi vs Old Delhi - I cannot remember where I stayed, but I chose something near to the train stations so I didn't need to haul my backpack too far a distance.
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