By far all the machines we have seen (even PDA, DFA, …) use states to "remember".

Given a TM that can hold a maximum of n states, can I conclude that it won't be able to remember more then n steps (for example won't be able to count more than n '0' symbols)?

You have to define what exactly you mean by "remember". In any a TM with, say, 100 states (the universal TM) can simulate

any other TM, regardless of the number of their states. So the answer to your question **in general** is no.

In order to simulate a TM the Universal TM needs to either have that TM as an input on the tape or write the TM to the tape itself, correct?

Isn't it true that simulating a TM with 100 states takes more than 100 states?

Just to add to Rani's answer, the universal TM (with, say, 100 states) can simulate any TM (say one tape, to make life technically easier) with **any** number of states. So the answer to your question ("Isn't it true that simulating a TM with 100 states takes more than 100 states?") is **NO**. Of course

to do this the universal TM does not attempt to remember the simulated TM in its state, but rather gets its description as part of the input, and

"consults" that description on every move.

Can I conclude from here that if I receive <M,w> I can build a TM that receives an output x and behaves exactly as M does on this x (accept/reject/loop) but has no more than 100 states?

It seems a bit odd since it's like saying that any TM with more than a certain number of states can be reduced to a TM with fewer states and both TMs receive the exact same language..

No, you can not conclude that.

What you can conclude that given <M,w> you can build a machine U that behaves like M on W.

The language of that U is exactly Atm.

What you need to understand is that anything U gets on one its tape at the beginning of the computation is **part of its input** and **affects the language it accepts**. You can not assume that the machine U has some <M> written somewhere and then it behaves like M for all inputs x. this indeed would mean that a TM can accept any language contrary to all that we have learned