Halting problem for finite tape TM If we have a primitive CPU/computer with small amount of registers and/or RAM, it is easy to check if the program will loop endlessly: just write down all registers/RAM cells states at each state and wait for repeating one.
If we reached repeating state, we got to know that the program will loop inifinitely and will never terminate.
We just need a CPU/computer with bigger RAM for the task.

If so, does it mean that halting problem exists only for TM with infinite tape/RAM, so it is not possible to write down all tape states?
"Easy to check" is the understatement of the century: can you actually carry out your proposed plan of "just" writing down all the registers/RAM cells, etc? You're right that it takes finite time, but a large amount of finite time. Our best algorithms for performing this check are exponential in the size of the input on the tape, and a strong version of the $P\neq NP$ conjecture states that there is no algorithm that achieves a faster (subexponential) runtime uniformly over all inputs.

So the P=NP problem is a finitary version of the halting problem. The latter has no algorithmic solution when input length is unbounded. The former admits a brute-force solution, but the amount of time required to find it is so large that for all practical purposes it's as good as unsolvable.

Edit: My statement that the halting problem can't be decided by any algorithm because the "input length is unbounded" is inaccurate, since for DFAs the latter also holds but the languages they accept are of course decidable. But certainly when the number of configurations is finite -- as is the case for a finite-tape TM -- a brute-force search solves the halting problem. Somewhat confusingly, the finiteness of the number of configurations is actually not necessary for decidability, as the DFA example shows.