I have 2 PSN accounts on a ps4 console. The US one has the PS+ subscription, the Asian one has not. Can I play online on the Asia which is not subscribed? Can I also gain trophies on the Asia?
Yes, both of them.
On PlayStation 4, if the "Main" account of the PS4 has PS+, all other accounts on that device (up to 16, no matter where they says they're located) can play online, and you can play any downloaded game on the device no matter with which account you bought/downloaded it with; i.e. I used my asian (secondary) account to download Ace Combat and play it on my US (primary/main) account.
If you use the asian account to play, you will earn trophies but in that account, not on the US one.
Other thing is, you can't copy the save file for a game from one account to another, well you can copy it to a usb when logged in with asian account for example, and then copy it back on the US account, but you can't use that save file to re-earn those trophies on the US account. I've already tried it, we laughed so hard when we saw a notification stating "You did not earn a trophy" when I tried lo unlock one.
You can't play Online without a Ps Plus account on the PS4 and you can't gain trophies on that account, without playing trough it.
If the game supports on-line play without PS+ then you can play it. It's up to the developers to require that. Some games allow on-line multiplayer without PS+ but with limited options.
Trophies are related to your profile that is playing the game and have nothing to do with PS+. If you play a US copy of a game on both profiles. Each profile will collect it's own set of trophies based on achievements completed while playing the game from that profile.