Can I pass a skill check even if my skill level is below the requirement?

From doing some reading here and actually playing through New Vegas currently, it would seem that no, these skill checks are absolute. Much the same way that you cannot pick certain locks without 60 Lockpick skill, the same applies to speech checks for things like Barter and Speech.

I have played an insane amount of NV and can confirm that there is no cheating around these, your options are raising your skills (I always max speech and lockpicking first in NV and FO3) and always grabbing every magasine you can (they have no weight so don't worry about taking too many), you can even get free ones in the Vegas with the mailbox like machines by the NCR building, (not the large embassy but the train station opposite from the topps.

I would recommend saving often on console, if you play PC just make a quicksave everynow and then (should be bound to F5) and if there is a check that you really want to do than revert your save. It's just one of the few annoying things about first time playthroughs.