Hi - so glad you are getting started composting! Best thing you can do for your garden!
1. Depends on where you are whether rats can be an issue or not. If you are in an urban area, where rats are a problem, then yes, they can get into your compost pile. Other critters can as well. Where I used to live (in the city, four miles from downtown) we had lots of raccoons. If allowed, they would tear the compost pile apart and drag stuff all around the yard. The answer to that is easy. You need some kind of tight bin or cage (NOT air tight, just animal proof).
Something like this but with a lid:
or one of the plastic ones (they come in many different versions)
Now that I live out in the country, but much less wooded, I have lots less critter problem and nothing has been disturbing my open three-sided bin.
2. Yes, you can have a compost pile in your garden. One way this is done is keyhole garden - Its a compost pile surrounded by garden.
Or you can just put a compost pile in a garden spot that you will plant next year.
My only concerns about compost pile in a garden, right next to your veggies, would be if you are putting manure in your compost pile. That's an okay thing to do, because ultimately the manure will be nicely broken down and just compost. But in the meantime, while fresh, it can contain pathogens. So I wouldn't want it right next to my veggies. But if your compost is just leaves and grass and kitchen scraps, then there doesn't need to be any distance.
3. People do make compost bins out of large plastic garbage bins. It needs to have LOTS of air holes. I still think it is trickier to get the balance right of air / water/ brown/ green doing it that way. It is likely to be a smaller pile that way and bigger piles actually work better. And if your pile is not sitting on the ground then, earth worms and beneficial soil microbes can't get into it and they do a lot of the breaking down. Personally I would take the bottom off of the garbage bin, so your pile can still sit on the ground, just have the bin around it for protection. If you don't do that, then you should add handfuls of good garden dirt as you add other stuff, and hopefully add some earthworms as well. You can't really tell in the picture, but the plastic compost bin pictured does not have a bottom.
4. Managed properly and working right, a compost pile does NOT stink (at all - has only a slight pleasant "earthy" smell). If your pile is smelling bad, something is wrong. Probably that means it either does not have enough air circulation or it does not have enough "browns" for how much "greens" there are, or some of both. (If you don't know what that means, please read the composting basics and composting 101 threads at the beginning of this Composting Forum.) Fix the problem by increasing the air circulation and mixing in more browns and the smell goes away.
5. I don't know pros and cons of burying a fish. But I know where I used to live with all the raccoons and other critters around, it would not have worked. The critters would have dug all the plants up to get to the fish. I know this because I tried fertilizing with fish emulsion. Even though the fish emulsion was diluted in a lot of water, still they tore up all my plants trying to find the fish.
hope this helps some. Keep us posted how it goes for you.