I have flights from Manila to Singapore and from Singapore to Bangkok, both on a different airline and ticket. Both flights will be in Terminal 1 and have a 5-hour gap in between. Do I still need to go through and exit immigration (for Manila/Singapore and enter again (for Singapore-Bangkok)?
Singapore has sterile transit available in all terminals, so if your bags are checked through or you have hand luggage only, you will not go through Customs or Immigration.
However, you may go through security on arrival at the gate, and you will go through security at your departure gate. Singapore laws apply if you're caught with any contraband, up to and including the death penalty for drug trafficking.