Ps3 – Does a game stay on the PS3 if I sign out

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.

As long as you never create a PSN ID for your other profile, there should be no problems and you will not have to purchase multiple copies. If you sign up for a PSN ID for that user, then you could still play with that DLC, but at the risk of violating the Terms of Service and possibly getting accounts banned.

Content downloaded onto an activated console is accessible to all users on the same console as long as the console is activated with the owner of the content's PSN ID. Some content specifically prohibits this.

Content is not shared in the store across PSN IDs and only the account that purchased the content will have the ability to install the content. To share content across accounts, simply have the owner of the content activate the console on their account and install the content on the same console.

For content purchased prior to November 18, 2011, you were allowed to have that content on up to 5 PS3s, and content purchased after November 18, 2011 is only allowed to be installed and activated on up to 2 PS3s.

Sharing across multiple PSN IDs violates the TOS and some users have been banned for it.