Helseth is House Hlaalu. Him being king of the whole Morrowind, does he get to decide over the important issues of the House Hlaalu's inner politics or not? For instance: would he have any word on who is proclaimed Hlaalu Hortator?

No i think he put someone in charge of the political stuff of hlaalu so he doesnt have to think about this stuff.

not likely; just because he's king doesn't mean he's in control of House hlaalu...granted he probabally has a lot of power in it, the Hlaalu Council probabally has more power.

On the Mainland Helseth would do so. Other than paying bucketfuls to send a boatload of DB to the Island, Helselth doesn't wield much power, it's Duke Dren who decides the hortator on Vvardenfell, and the Hlaalu power.

But is Hlaalu Council bound to do what he tells them? I mean does King of Morrowind get to interfere in internal affairs of a Great House, even if it is his own House? Is he a member of a leader of House Hlaalu?

No I dont think he have such great powers within House Hlaalu, at least not in the Vvardenfell district.

Helseth would be allowed to do what he wishes with House Hlaalu, provided he has enough followers in enough places to do so. I think, if the King began steering towards policies which would endanger or diminish House profits, then expect for some the merchant princes within the House to start making plans for a replacement.

On paper, the King of Morrowind should be allowed to "interfere" with other Houses, it's not like that in reality. Things will get done when there are enough people in the right place to get them done, therefore, the King or Hlaalu has little influence in the other Great House provinces, like Telvannis.

The King's interfering persistently, and too drastically caused Hlaalu, Redoran, Dres and Indoril to all go into civil war with one another.

King Lethan was Hlaalu. The nobles in the city were angered over their man being gone when Helseth took the throne, but the commoners clearly didn't care too much (they don't like western kings). The only one who did was the common tongue printer and he was related to the widow of the King I believe, so the assassination outrage realy fell on deaf ears.