However, someone recently told me that in the rulebook it says that if you play a blood angels successor chapter, the successor chapters name keyword ("flesh tearers" keyword) would replace the blood angels keyword in all instances.
Technically what you were told is correct, every unit in a detachment needs to have the same faction keyword.
However, this does not mean that your Flesh Tearers force needs to have the Flesh Tearers faction keyword. If you take a unit/character that automatically has it (like Gabriel Seth) then yes you would need to make the swap for the rest of the units in that detachment, but otherwise you can run your Flesh Tearers with the Blood Angels faction keyword and be perfectly fine (to my knowledge you don't gain any additional rules in this case.)
but he also continued to tell me that because they then have the flesh tearers keyword and no longer the blood angels keyword, my army isnt allowed to use any of the blood angels relics in the BA codex, any of the characters including Mephiston or anything of the sort. because they all tend to say "if a blood angels keyword unit..." on them.
This is correct, the relics of Baal are limited to a detachment(s) with the Blood Angel faction, same goes for named Blood Angel characters like Mephiston.
Again though, if your not taking a unit like Seth, you can keep the Blood Angel keyword and be about your merry way.
Alternatively, if you are taking Seth, you can run two detachments with one of them having the Blood Angel faction (though this would mean they could not benefit from any faction specific bonus's granted by Seth._
Surely just because I decide to have my guys as a successor chapter, it surely cant mean that im no longer allowed to use half the books worth of content?? It would be really dissapointing if I were to discover that because iv painted my guys to be this cool subfaction, I literally loose almost everything that assists this chapter in doing well in games.
As Dave and I have pointed out, you would be limited in named characters and relics if all of your detachments use the Flesh Tearer faction; but there is nothing preventing you from running one or all of your detachments with the Blood Angels faction (unless your fielding Seth of course.)