I have one more question, is there any cryptocurrency whichs effectiveness, speed, trust is somehow related to bandwidth speed?
There used to be a coin that can be obtained using your bandwidth, but because of the limited high-speed internet available all over the world, it didn't end up too well for the coin and the devs. Also, not all high-speed internet plans nowadays are unlimited (which is a shame), so there's that.
On some low-difficulty altcoins which utilizes GPU power, yes you can, though there aren't that many coins that are virgins on GPU miners, so your chances of finding one is very slim. In order to get some kind of decent profit in mining, you must obtain at least two or three GPUs and mine whichever is profitable. It might not make you a lot of money but everyone has to start somewhere.