So PVP and PVE skills/items and heros

If i use bath fac in PVP to unlock new skills or items or heros do they get unlocked in PVE?

no if u use balthazar faction to buy a skill u won't have that skill on PvE but if you have an Hero on PvE that hero will get that skill.

also wrong section, this should be in Questions & Answers

If you buy skills then you can either buy some in pve at skill trainers a bit earlier than usual but i general they can be got via normal tomes for that profession , elite skills can only be got via elite tome or by normal method of capping.
Any skills unlocked via balthazar will be readily available on your heros in pve but sadly any pvp hero unlocked stays as pvp and wont be accessable in pve untill you unlock them via normal means.

There's a difference between unlocking and learning/obtaining. Unlocked skills can only PvP characters and heroes use, and unlocked items can just PvP characters use.

However, if a skill is unlocked, your PvE characters can learn that skill more easily than normal. This is because unlocked skills are available to every skill trainer (from the right campaign) and you can use tomes to learn them.
Elite skills, can only be learned from capturing, or if they are unlocked, by the use of an elite tome.