My question is: If House Hlaalu is no longer on the council. What happened to King Hlaalu Helseth and Barenziah? Did they relocate to the new capital? or Where they ousted by House Redoran?

I ask these questions because like his name suggests. Helseth was a member of House Hlaalu.

Probably killed. Tons of bad stuff happened to Morrowind, and I don't imagine any "Imperial puppets" surviving.

well barenziah died of old age if I remember correctly and a snake like helseth makes a lot of enemies so he probably met his end at a morag tong blade.

I believe Mournhold was sacked by the Argonians. It's possible they were killed then.