I haven't booked the flight to Goa yet, I feel the need to plan the domestic journey first, then book.
Okay, sure. In which case I'll re-iterate the possible acclimitisation benefits of getting off your international flight and onto a flight to Leh. Worth considering I think.
Are there any other parts of the plan that don't make a lot of sense to you?
Well, as above - your plan for Rajasthan of Udaipur / Pushkar / Jodhpur doesn't make sense either. Much better better if you go Udaipur - Jodhpur - Pushkar then to Agra.
If you wanted to get to Kalka (and you'd only do that for the train ride to Shimla) then there is a station called Tundla (large station, about 45 minutes outside Agra) - it has a daily 12 hour train to Kalka - roughly 16:00 to 04:00 - you could sleep for a fair portion of this. Tundla is a good option for a train to Haridwar too - leaves roughly 01:30 and arrives at 11:00.
I'm still luke warm about Goa, and it isn't a great fit with Hampi if you plan to take a train. Taxi or bus is a different matter. You could probably most easily get an international flight into BLR, go immediately to Hampi and then take an overnight train to Hyderabad for a couple of days.
https://www.lonelyplanet.com/india/andhra-pradesh/hyderabad-and-secunderabad
And you'd probably still get a flight HYD-UDR (via DEL) inside 5 hours and inside 5,000 INR.
The other thing you might want to consider is finishing a given leg (or module) of your travel near an airport to set you up for the next leg. Rishikesh is a good example of this. There is an airport at Dehradun, the airport is barely 30 minutes by car from Rishikesh, and the flight to Delhi is just an hour.
For example I'm thinking of cutting the Uttarakhand leg of the journey because I don't think there's a way from Uttarakhand to Himiachal Pradesh without heading back the way I came.
Ah, that's not the case at all. From Haridwar take a train to either Ambala or Chandigarh and then to Kalka.
For example:
HW SGNR EXP (14711)
Departs:HARIDWAR JN (HW) - 14:15
Arrives:AMBALA CANT JN (UMB) - 17:30
Transit wait: 2 hour(s) 28 min(s)
Train: KALKA SHTBDI (12005)
Departs:AMBALA CANT JN (UMB) - 19:58
Arrives:KALKA (KLK) - 21:20
That's just 7 hours, and those trains run 7 days. Get a proper night's sleep in Kalka, and then take the Himalayan Queen to Shimla at midday.
The thing with travel in India, is that there are usually at least three viable options for getting from A to B. Your first task is being 'fixed' about where you want to go, and then figuring out the logistics.
Sing out with other questions.