1. Because you will probably be truncating the output to a short finite hash, and because each block of internal state only leads in one direction (it affects all later ("up") blocks of internal state for all later ("right") message blocks, but not previous ("left" and "down")), you can optimize it by only implementing the part that you need. So, yes, an actual implementation will be limited, but if you have enough memory and enough time then the limit can be as high as you want to be. (Maybe a diagram might explain it better; I am not sure that this explanation is any good.) (SHAKE also has infinite output (but finite internal state), and when using SHAKE also you would truncate the output to a finite size.)
2. A larger output size might risk leaking data, although I would think it would be difficult.
3A. Like I described, it then needs O(1MB) space and O(1GB) time, so it will be slow. However, I do not expect you should need a hash that long.
3B. I don't know; probably about as predictable as any random number generator, if the hash is designed correctly. (I only describe a construction, and the hash algorithm design involves more than that.)
2. A larger output size might risk leaking data, although I would think it would be difficult.
3A. Like I described, it then needs O(1MB) space and O(1GB) time, so it will be slow. However, I do not expect you should need a hash that long.
3B. I don't know; probably about as predictable as any random number generator, if the hash is designed correctly. (I only describe a construction, and the hash algorithm design involves more than that.)