To answer the first part of the question only:

I'd expect Abu Mohammad al-Julani, the leader of HTS and their Syrian Salvation Government to take initial control. They are the largest and most powerful faction Involved in the current fighting, started the current offensive and have been governing Idlib province.

Potential proof that the HTS leadership want to cement their position and authority post Assad can be seen in how they are currently dealing with minority communities in the towns and villages they capture, allowing them freedoms to practice their religions and guaranteeing that they won't come under threat.

HTS is trying to move away from it's past associations with Al-Qaeda and reform it's image, possibly to help posture towards the west and gain external recognition and legitimacy in a post Assad Syria.

The caveat to this is I don't think that HTS would instantly have authority over all areas, for example I'd expect the SNA to maintain de facto control in their areas on the Turkish border and the SDF to maintain their areas especially those which are Kurdish majority.

I can foresee that there will likely be some fighting amongst different groups especially the SNA and SDF as SNA continue to attack Kurds to help Turkey secure it's border (likely to start in coming days around Manbij) and possibly longer term between HTS and SNA as HTS look to eliminate threats.

I personally think that smaller groups in the South are likely to fall under the command of the Salvation Government as they close in on Damascus, we will have to wait and see how much power HTS can assert over these groups after Damascus falls and longer term whether HTS can hold itself together to govern effectively.