I have a 2005 Rx8 and it has hot start issues and the compression test showed the lower side of normal. Who is good to do replacement valves in Auckland?

 

Wow.... valves ;-(

 

Um, no valves in this engine...

 

What you need to do is look for a rotary engine builder in or near your area. The average corner garage will be totally lost with a rotary engine and,if they even try to rebuild it, will proably make more of a mess than you seem to already have. The alternative, though more expensive, would be a Mazda dealer and one of their factory rebuilts. If you go that route, though, budget an extra $300. to $400 for extras like new ignition(possibly the root cause of the failure) and coolant flush and fill as factory rebuilts often have excess sealant floating around in them that can clog the rad core.

Also, as a sideline, you might want to have a look here for someone local that can help you out with some advice on who is good to call:
https://www.rx8club.com/australia-new-zealand-forum-37/