For example, when it had been vacant for a long time, why was the title of king of Bohemia still in existence in 1618? And why was there no king of Austria, if you could have kings in the HRE? Who kept these rules, or enforced them?
I'm linking to a previous answer i made about how would a medieval king give more land to a lord. https://reddit.com/r/AskHistorians/comments/z9yhej/how_would_a_medieval_king_give_more_land_to_a_lord/iyomvl5
The Emperor has the authority to give and take titles, manage vacancies and raise nobility to higher titles. BTW, I don't know what you're referring to the Kingdom of Bohemia being vacant until 1618. The Habsburgs ruled Bohemia.
The Emperor could and did raise a Duke to a king. The Dukes of Bohemia were raised by the Emperor to the non hereditary title of king a few times, meaning that their sons wouldn't be king but dukes again. Bohemia wouldn't gain permanent kingship until Ottokar I was able to negotiate himself being raised to King with Philip of Swabia King of Germany (but not Emperor) during his struggle against the Anti-King Otto IV.
After Philip was murdered, Otto IV also recognized Ottokar as King as did the papacy. With the final recognition coming from Emperor Frederick II who signed the Golden Bull of Sicily of 1212 which officially raised it to a kingdom and freed Bohemia from all feudal obligations except for sitting on imperial councils.
Bohemia would remain an important part of the Empire with several Bohemian kings becoming Emperor, becoming one of the few Electors in 1356. And several Emperors becoming kings of Bohemia.
What we can see is that there's nothing really stopping the Emperor from raising a duke to king. But there's also the Papacy who can have a say, and who also creates kingdoms, as well as the need to be able to defend your title. Otto IV and Frederick II had opportunities to revoke the kingdom until it was sealed by several charters and Golden Bulls.
Now, as to why Austria wasn't a kingdom. Austria started life as a small margrave (border county) within Bavaria which was only the small area around Vienna. Over the centuries, various emperors would grant it independence from Bavaria, raised it to a Duchy, and of course granted it to a new family.
The Habsburgs were swabian counts who were granted the duchies of Austria & Styria and became King of Germany at the same time, battled the Kingdom of Bohemia for these territories, it was a big game of thrones we won't get into.
The Habsburgs gathered a few more adjoining duchies to Austria and Styria from which their powerbase would remain until 1918.
In 1356, Emperor Charles IV of Luxembourg proclaimed the Golden Bull of 1356 which put into law the 7 electors of the empire who also received many rights & privileges. Prior to this all the nobility had a vote and things got messy from time to time.
Importantly, the Habsburgs of Austria were not electors. Soon after the Golden Bull of 1356, the Habsburgs would forge the charter Privilegium Maius which gave Austria the same privileges as the Electorates, except the vote, and raised the duke of Austria into the Archduke of Austria with their other duchies included into the Archduchy.
"Archduke" was basically made up by the Habsburgs. There's one reference to an "archduke" prior to this except it's a bit muddy to what it meant by that. And the Habsburgs just made it up. There's only one archduchy in history. It's just Austria.
Now, the Habsburgs started using the title but the Emperor didn't recognize it; it's a real "established titles" moment.
It wouldn't be until Emperor Frederick III, a Habsburg, that the archduchy was officially recognized. Think the Obama giving the award to Obama meme.
Why didn't he say he was king of Austria? I don't know. I never saw any references to them ever even wanting to, not that there was something restricting the ambition.
The primary kingdoms of the Holy Roman Empire were the Kingdom of Germany, the Kingdom of Italy, the Kingdom of Arles (which would be mainly lost by inheritance to the French) the Kingdom of Bohemia.
Later the Emperor Leopold I would recognize the Duke of Prussia Frederick as King in Prussia in 1701. This was recognizing Frederick as king of his territories outside the Empire but not of his other territories within the Empire.
The final Kingdoms the Holy Roman Emperor would create were the Kingdoms of Württemberg and Bavaria during the Napoleonic wars. We give credit usually to Napoleon for doing this but it's actually the legal authority of the Emperor who did this. But because Württemberg and Bavaria sided with Napoleon against the Habsburgs.
It didn't matter because the Holy Roman Empire would be dissolved the next year in 1806.