The President of the UN Security Council's role is as follows:
The role of president of the Security Council involves calling the meetings thereof, approving the provisional agenda (proposed by the Secretary-General), presiding at its meetings, and overseeing any crisis. The president is authorized to issue both Presidential Statements (subject to consensus among Council members) and notes, which are used to make declarations of intent that the full Security Council can then pursue. The President also usually speaks to the press on behalf of the Security Council.
Wiki citations 4 to 8 in the above are have broken links, but they're references to the Provisional Rules of Procedure of the United Nations Security Council (Chapter I, Rules 1 and 2; Chapter II, Rule 7; Chapter IV, Rule 19), which you can read in pdf format here, and examples of Presidential statements and notes.
More specifically, is he who decides which countries will speak and in which order? What happens if a representative of a country interrupts another country, and what would the president do in such a situation? Is he who decides what happens if the rules aren't respected?
All of those are what the presiding bit in the role are about in the above. The Security Council's rules seem to cover some of your questions explicitly:
Rule 27 covers who gets to talk:
The President shall call upon representatives in the order in which they signify their desire to speak.
Rule 30 covers points of order:
If a representative raises a point of order, the President shall immediately state his ruling. If it is challenged, the President shall submit his ruling to the Security Council for immediate decision and it shall stand unless overruled.
Rule 33 covers a few more points related to interruptions:
The following motions shall have precedence in the order named over all principal motions and draft resolutions relative to the subject before the meeting:
- To suspend the meeting;
- To adjourn the meeting;
- To adjourn the meeting to a certain day or hour;
- To refer any matter to a committee, to the Secretary-General or to a rapporteur;
- To postpone discussion of the question to a certain day or indefinitely; or
- To introduce an amendment.
Any motion for the suspension or for the simple adjournment of the meeting shall be decided without debate.
I haven't found anything in my cursory scan, besides rules 30 and 33 above, with respect to what happens when members are unruly and contentious. That being said, the SC is small enough that I'd suggest they don't need a Bercow around to keep debates orderly.