Just to add some detail. You probably know that the flapper is the pink thing. It's a Korky flapper. Korky has a set of instructions on how to install it and I suggest you review it and follow it.
http://www.korky.com/PDF/instructions/2000BP.pdf
What you will learn is that (1) you gave it way too much chain, so the chain is hooking under the flapper, preventing it from sealing, and preventing the tank from refilling; (2) When you installed the flapper, you bought a good universal Korky one. Flappers fit two types of flush valves (the flush valve being the thing the flapper sits on with the big vertical tube) those with the little wings/hooks on the side (usually plastic) and those without (usually metal). The universal one you bought works on both types, but has to be adjusted to work on one type or the other. If you had no wings on your flush valve, you would have hooked the circle ("collar") over the tube, as you have done, but there would be no hooks on which to attach the flapper. Because you have hooks/wings on your flush valve, you need to REMOVE the circle (collar) from the middle, as shown in the Korky instructions, and JUST hook the flapper to the hooks/wings. All it takes is a pair of scissors. Now the flapper will fit properly on your flush valve and will move up and down properly.
You should leave it, when it's resting, with a link (or two at the most), flopped on the top of the flapper. It needs a little bit of chain so you don't inadvertently keep it from making a seal. But no more than two links. That should also solve your "holding the handle down" problem. If it doesn't open itself and stay open with 2 links, then make it one, and that should fix it.
If you continue to have filling issues, you can replace that fill valve with a Korky 528MP (the one with a silver cap) Fill Valve, available at Lowe's and a lot of other stores. It should take 10 minutes, with minimal tools. It's a very easy DIY project. However, you probably don't need it; just fix how you have installed the flapper.