A few food questions

This spring I will be visiting Tokyo, Kyoto, Hiroshima, Hakone, Kanazawa, Takayama, and Miyajima. In any of those places:

1) Where I can try horse meat?

2) What are some recommended sushi buffets up to $20/person?

3) I heard that Japan has really strange ice cream flavors. What is a good place to find some of these? Is there an ice cream shop that's known for weird flavors?

1) If you are in Kyoto or Tokyo, you can try horse meat, "ba-sashi", at this restaurant: http://r.gnavi.co.jp/c234200/

There are others, but I think this one might win on flavor and experience.

2) For inexpensive but good sushi in Kyoto, go to http://www.hotpepper.jp/strJ000019998/.
It's in Pontocho just a 5 minute walk from Sanjo on the Keihan line.
In Tokyo, Sushi Zanmai is pretty good and they sometimes do live "kaitai" performances where they cut up a 300 lb tuna and then offer fresh sushi.

Japan does have lots of strange ice cream but the strangest ones like jellyfish will not likely be available in the cities. You should be able to find red bean, green tea, soy etc. near temples, etc.

What you should try is the caramel ice cream from Hanabatake Farms. http://www.hanabatakebokujo.com/special/hot-caramel/index.php

They have multiple locations in Tokyo.

That caramel ice cream looks delicious, I will definitely try it.

Regarding sushi - I was actually hoping to find some all-you-can-eat sushi places in those cities preferably up to $20/person.
Do those places offer all-you-can-eat, or are there other such places?

I'd also like to try Kobe beef, but it's far out of my budget. Would it be viable for me to buy some from a market/store and prepare it by myself?

The best meals are always the ones you prepare yourself!

If you plan ahead and have access to some sort of mini kitchen, you can buy fresh luxury ingredients from specialty shops (butchers, fish markets, farms) for much cheaper than what you'd pay at restaurants.

The best beef in the world, Tajima Beef, can be bought online and many producers have specialty shops for less than $50/lb (450 grams).

Just for perspective, a pound of Tajima beef (or Kobe, etc.) is considered a serving for 2.

There are places with sushi "baiking" (all you can eat) for $20-30 during lunch time but I can never recommend all you can eat buffets. You are getting a collection of dinner menu leftovers, nothing more.

Sushi tetsu mentioned above has a great selection of $1 sushi and the flavor is good considering the price.

2) ''All you can eat'' and ''good sushi'' simply don't go together :)

3) Correct me if I'm wrong, but I believe that this ''Ice Cream City'' located in Namco Nanja Town in Sunshine City (Ikebukuro, Tokyo) is open all-year-round:

http://www.namja.jp/ice_cream/event/bn/event09.html

I don't know for horse meat, although I've come across it a few times...

Cheap sushi are usually not of very good quality, so if you want to eat good quality, I suggest you don't choose an all-you-can-eat place...

As for weird Ice creams, Macha (green tea) flavor is common. There are also a lot of Azuki (red beans) ice creams around Asakusa, and I suppose other touristic spots too.
But as AK said, Ice Cream City in Ikebukuro has lots of flavours, like salmon, noodles, or rice...