What should the arbiter and what should have I done in this case? Recently, an official (FIDE-rated) tournament was held, where I participated.
In the last round – which was the most critical one – I was black and my opponent white. In the middle of the game, they started making strange sounds, which clearly disturbed both the nearby boards and me.
At first, I mentioned that they were disturbing. Since they wouldn't listen, I called the arbiter. In all 4 cases, the arbiter warned the user like this:

&lt;Player name&gt;, could you please stop doing this, since you are disturbing your opponent?

In addition, when I offered draw, here's what my opponent replied:

If the game were drawn, then I would offer it first to you.


What should the arbiter have done? At least, I'd expect to have a punishment in the clock (e.g., +2 minutes for me, -2 for them). It is well-defined in article 11 of Fide Laws of Chess (emphasis mine):


11.5 It is forbidden to distract or annoy the opponent in any manner whatsoever. This includes unreasonable claims, unreasonable offers of a draw, or the introduction of a source of noise into the playing area.
11.6 Infraction of any part of Articles 11.1 – 11.5 shall lead to penalties in accordance with Article 12.9.
11.7 Persistent refusal by a player to comply with the Laws of Chess shall be penalised by loss of the game. The arbiter shall decide the score of the opponent.

And I suppose that this behaviour is persistent.

What should I have done? I ended up losing the game and, while I could end being 2nd, I ended up being 3rd! Could I make an appeal to another arbiter or even the chief?
First, we cannot tell you what the arbiter should have done because we were not there and certainly don't have all the facts. We only have your version of events. What you say was very disturbing may not have been perceived as so disturbing to others there at the time. We weren't there, so we just don't know. We can't judge.

As to what you should have done, you did the right thing by bringing the disturbance you suffered to the attention of the arbiter. If the arbiter took no action beyond issuing a warning you could have made another appeal to the arbiter during the game for more robust action. 

At the end of the game you could also have refused to sign the scoresheet and instead raised the issue again first with the arbiter and then the chief arbiter. If this failed then you could have lodged an appeal with the appeals committee for the tournament. FIDE rated tournaments are required to have an appeals committee to which you can appeal during the tournament. This committee is usually composed of players in the tournament and disbands when the tournament finishes.

Finally, if you took all those actions and still did not get a satisfactory answer you could appeal to your national chess federation. You could probably still do that now but if you didn't make the previous levels of appeal and complaint you are unlikely to get any satisfaction.

Currently there is a high level women's tournament (candidates for world championship) taking place in Kazan in Russia which has a problem with noise from a band playing music outside. Some of the players have earplugs and are not being disturbed by it others do not and are being disturbed.