Ps3 – Does R1 game update in PSN Asia account

You've over complicated this.

For PS3, basically once a DLC was installed on a system, any user on that system can play that DLC and earn trophies for it, whether signed in to PSN or not, I've never seen DLC which is only usable by the buying user except for items like Home items, Avatars and DLC for one specific game I know of.

So your "process" is fairly simple:

1. Log into the appropriate store for your game's region (you'll need another PS3 profile with a PSN account in that region)
2. Buy DLC from that store (funds could be from a local credit card or prepaid PSN card) and install it.
3. Log back into your main profile and account and play the DLC.

PSVita is different. A PSVita system and memory card is locked to a single user, so even if you switched the Vita's user (system restore), you'd still be only able to download the DLC to a memory card assigned to the "foreign user", meaning it won't be accessible to your "main user" (which will need its own memory card), this means that in order to buy DLC for a PSVita game, you must own a copy of the game from the same region as your main PSN user. There's no current way to avoid this (at least one that I know of), and it basically means any PSVita game that has "required DLC" like online passes, UPlay pass, etc, will only be "fully playable" if you buy games from your own region.

The short answer is No. You probably will not earn trophies to your account

In the situation you've described, where you use a save on an account where it wasn't created, will almost 100% make trophy earning ineligible, even if you copy it back to your system afterwards.

Your best option is to create another profile on your friend's system and log in to your PSN account, and on that account use your saves, this way you can ensure trophies will unlock. Make sure to sync your trophies when you log on to your profile for the first time and before leaving your friends house, so anything unlocked during your stay will transfer back to your account.