Rail pass for Tokyo to Hokkaido

I understand there are some special shinkansen not eligible with the rail pass, which means going to Kyoto from Tokyo takes 3 hours instead of just over 2, which is fine.
However I have heard conflicting things as to whether all hokkaido shinkansen is covered. On this and some websites says yes it is covered, but then some people have cautioned me that I should double check that since some express services may not be.

I plan to take the Tohoku-Hokkaido shinkansen from Shin-Hakodate station to Tokyo Station, then a few days later use the train pass for a round trip from Tokyo to Kyoto, all within a week. I will be flying to Sapporo so getting to Hokkaido is not a problem, coming down to Tokyo is the issue (one-way).
Shin-Hakodate to Tokyo is meant to take approx 4.5hrs which is fine, but if we have to take slower trains and it becomes 6 hours that may be a problem.

You know this is a Life time visit.It means millions of people come to japan only once in their whole life time.so in this case you will buy JR pass 7 day=29650 yen cover whole japan including tokyo to hokkaido as well as hokkaido shinkansen.

But as a student i cant buy JR nationwide pass ,so i will buy 18 kippu=12050 yen cover only JR local train whole of japan

The Hokkaido shinkansen is covered with the nation wide pass.

I have never read differently ever.

Various regional or local passes exclude it. But these passes arent the ones you are asking about.

As mentioned by the others, all trains along the Hokkaido Shinkansen are covered by the Japan Rail Pass. The only shinkansen that are not covered are the Nozomi and Mizuho along the Tokaido/Sanyo/Kyushu Shinkansen.

More details can be found here:
https://www.japan-guide.com/e/e2361.html

Thanks, got all the info I needed about Hokkaido, in short, it works.
About the Kyoto part however, since I have to take the less frequent train, just how frequent is it? Is there still one eligible train from Tokyo every hour, or is it much less frequent than that? 3 hour trip up from 2 is fine with me.

Every 30 minutes

http://www.hyperdia.com/en/ for your train times

Sure Shin-Hakodate to Tokyo is typically around four and a half hours (only four going north), but Sapporo to Shin-Hakodate Hokuto is almost four hours by itself. Sapporo to Tokyo by train is eight and a half hours - that's why people take a cheaper and faster option and fly.

I understand train to sapporo is not very practical, but I am only taking train one way back from Hakodate.
I will be flying to Sapporo and then slowly taking bus/trains down to stop at various places along the way, spending a few nights at Sapporo with daytrips out of there to the northeast/interior, then a few nights in Otaru, Niseko, Toya and Hakodate. So I plan to buy my train pass when I am ready to leave Hakodate and then use it to get to Tokyo. To me taking the train under the sea and then the scenery on the train all the way down to Tokyo is part of the fun :)
Worth since after resting a day in Tokyo I will be moving on to Kyoto and then back to Tokyo within the 7 days.
So it's something like 7 day pass:
Day 1 Hakodate>Tokyo
Day 2 Rest day, use for tokyo metro
Day 3 Tokyo > Kyoto
Day 4+5+6 In Kyoto, may be able to use it for local lines but if not eligible not a big deal
Day 7 Kyoto > Tokyo
Seems like a very good deal.

If you are starting Shin-Hakodate-Hokuto, then, sure it makes sense and there are some good savings for the three main trips you plan. But you said you were going to Sapporo, which is a long way from Hakodate (in time), namely four hours by train. Not sure of time required for the bus/road, but you have other spots to visit as well.

Cannot use JR rail pass for Tokyo metro subway.