Not sure there is precedent for any of that, given that this is supposed to be more of a ceremonial role. So before the parliament elects a president the oldest or most senior member of parliament takes that role. So that person has neither a democratic mandate nor much power and is rather there as a master of ceremonies as someone needs to organize that and the older people usually have the most experience with protocols and and whatnot.
It is probably within his power to shut down hecklers and MPs overstepping the rules, but shutting down debates and proposals is actively sabotaging the work of parliament and the rights of the MPs which basically brought the inaugural session to a halt.
So according to a lawyer on constitutional matters he has definitely overstep his powers (german link), but might also have further violated the constitutional rights of the MPs. There are 2 opposing views on whether the self-organizing principle of parliament works before or after it's official constitution, but sabotaging it entirely is apparently massively overstepping the powers of an almost purely ceremonial position that does not have the same powers as an actual president of parliament.
Now how the court will decide will be seen, probably today already as they requested a speedy trial and the inaugural session is supposed to continue tomorrow so maybe just wait on how that will turn out.
And in terms of consequences well the minister of the interior already floated the proposal to ban the AfD (german link). Usually in that regard one distinguishes between merely holding views that are anti-constitutional views, which leads to further investigation, but is usually still covered by free speech and behavior that is actively aiming to work against the constitution. Where in this case sabotaging parliament probably adds a new bullet point to that list of evidence.
Also the AfD does not hold a majority in parliament so the rest could just elect a new president which they are supposed to do in the inaugural session anyway and he's definitely ramping up fines for violating laws of parliament but may even incriminate himself much further, but not a lawyer and we're likely going to see the results of that very soon anyway.
Update: Apparently the Thüringer Verfassungsgericht confirmed most of the points of the CDU, which includes the legality of what they were proposing as well as the autonomy of parliament to change the agenda for the day, to check if enough MPs are present to make changes and to change the organization of the election for president of parliament ahead of the election.
The point where they lost is to bind the Alterspräsident to hold a vote on the change of the rules of procedure, but apparently that's more of a technical issue, because so far they haven't gotten to the point where parliament had put that on the agenda and the MPs showed the will to do so, but as now the ability of parliament is reaffirmed and as there is apparently a majority, that shouldn't be necessary in the first place.
And they apparently gave the Alterspräsident a roadmap of how the next session is supposed to look like.
And apparently Treutner has acknowledged the ruling and promised to apply it.
Here are German links for the ruling by a legal blog:
https://www.lto.de/recht/nachrichten/n/nach-eskalation-im-thueringer-landtag-verfassungsgerichtshof-vgh-gibt-cdu-antragg-statt
As well as a live tracker for the session and the scandals around it:
https://www.mdr.de/nachrichten/thueringen/wahlen-politik/landtag-aktuell-live-ticker-konstituierung-start-sitzung-104.html?trk=public_post_comment-text