The two most important perks for hardcoe mode are:
Strong Back - Good for either mode, but I'd consider it vital for hardcoe.
Pack rat - Useless in normal mode, but even more vital then the other one in hardcoe.
Ammo weight was always my biggest problem in hardcoe and Pack Rat goes a long way to alleviating that issue.
I personally disagree on this one, I never use these perks. But I have to say my chars all have strength between 7-8 and 9-10 at the end of the game so maybe that's why I don't need them. I never have ammo weight problems tbh, if anything I have problems with the stuff I loot in the wastes, but companions come in handy as well. I usually have ED-E carry some backup ammo.
There are really no most important perks but there are handy ones, the ones above I can imagine being handy for players who go solo or have low strength. I always find light-step comes in handy, especially in the Dead Money DLC cuz u'll get crippled a lot otherwise and it's a pain to get ur hands on doctor's bags/hydra's. Also, on that subject, never throw away Hydra's cuz they can restore ur limbs and don't weigh anything. U can get addicted but this only happend once to me and if u carry a fixer it's never a problem. Give a couple of Doctor's Bags to ED-E as a backup and u won't have any criple problems again.
Finally I'd say to try and use sneak attack criticals, one bullet can really hurt or instantly kill an enemy that way which helps towards conserving ammo and increasing the time u can wander before going back/to a merchant to get more ammo.