Is it possible to get Mayers to Primm sheriff AND get NCR to raid Correctional Facility? Yes, I mean getting both of them done. If thats possible how can I do that?

I did it.

Lets see...

1. Do 2 quests for Eddie (Chavez + merchant).
2. Take Eddie's 3rd quest.
3. Go to Primm and rescue Beagle + clean Primm + hotel where Beagle was.
4. Talk to Beagle (or someone else) about new sheriff.
5. Go to NCRCF and talk to Mayers.
6. Get pardon for Mayers
7. Talk to NCR and betray Eddie.
8. Raid NCRCF

Is this correct order?

Yes you can, i did it in one of my playthroughs, Looks good. And make sure to get in the habbit of always making a Save game at certain points in general, unsure or not.

You can also do it by simply clearing Primm, heading up to talk to Meyers then getting his pardon, and worrying about the entire NCRCF line after he is pardoned and in Primm. The only parts that can block your progress are if you do the raid first, Meyers can get bugged and stuck in a loop only saying "goodbye".

I thought that you get vilified by PG if you clean Primm and thus Eddie will not give you any quests?

Nope, the only way to get Vilified is doing the gunfight quest in Goodsprings and actually hitting 2+ of the gangers raiding, or attacking any of the gangers you run into between Primm, Goodsprings and the NCRCF - and even then it's possible to avoid the faction hit if you successfully sneak-crit them before anyone notices you. All the convicts in Primm are factionless.