Note: These bugs have been happening to me from befoer the 1.4 patch.

1: I found all Unusual gems before speaking to Vex and the rest of the quest never trigers, the menu has 24/24 gems but nothig more hapens.

2: Helm of Witerhold in yngol barrow, What is suposed to hapen is theres a skeleton on a throne and you have fight its ghost then take a stell horned helm from a chest and a gate opens so you cn exit the dungeon. I started this quest at level 70 and reatemted at higher levels and instead of a skeleto and ghost theres a high level droug death lord that has some blue light streming out of him after killed and when you take the helm the gate neve opens on whene you return the helm to winterholm the quest compleates but you get no reward and the helm is stuck in the inventorie. (have never attenpted this quest at low levels).

3: I have played for the empire with two different characters now and after compleating the civil war quest both times Hogvar will become a permanet follower in whitrun. You cannot get rid of him as he is unkillable, he always has his weapon out and get so close to you that he practically murges and its anoying.

4: All bard quest items neve leave menu (Finns lute, Panteas flute, Rjorns drum). Finns Lute has never worked on any character I have used.

5: Are the Attonment Spheres and the Runed Lexington ever suposed to leave you inventorie as for me in any playthrow they never have.

After deciphering your post, it's a bug that you collected all 24 gems before talking to Vex, reload a save. The helm is in a random dungeon, so can't help you there. Hadvar as a follower is a known bug but it should eventually fix itself. Make sure your talking to the right person at the Bard's College. Not sure on the sphere and lexicon, but you should be able to take them back to Septimus for Discerning the Transmundane.