When are stock prices updated?

In normal situation when there are no errors with the cron jobs:

SGX will be updated around 6-7pm SGT

HKEX will be updated around 6-7pm SGT

KLSE will be updated around 9-10pm SGT

USX will be updated around 7-8am SGT

Does the market price changing according to the current exchange rate simultaneously?

Not exactly sure what you meant but I use exchange rates based on ECB

https://stocks.cafe/exchangerate?exchange=XSES