Couple of suggestions (assuming you're in a group with players in a similar gear, and assuming the group does not consist of top server players)^
Lost Island (HM) (Tionese) - I'd recommend Columi
Eternity Vault (HM/NM) (Tionese/Columi)
Karagga's Palace (HM/NM) (Tionese/Columi) - Please separate HM and NM for these two ops. They're of completely different difficulty. For KP NiM you'd better to be in at least 50/50 Columi/Rakata. EV is a bit easier, but still, no Tionese for sure.
Explosive Conflict (HM/NM) (Rakata) - I'd recommend 50/50 Rakata/BH for HM
Sorry if this is sort of a stupid question... new player here. Can we bring pvp gear to hard modes?
The general rule is that PvP gear is 1-2 tiers down on PvE gear based on stats. So if your PvP hear has better stats than PvE gear (discarding Expertise of course), you're good.
As said above, there are some players who zealously object to using PvP gear in PvE even if PvP gear has better stats. Thankfully such cases are not common on my server.