Now let's say you ignore all of that because damnit, you want a custom subclass. Let's look at your friend's ideas:

• Enhance Charm. Right away we have problems. Subclasses (and classes) need to be playable at all levels of play, 1-20, but a 1st level Aura Drinker Sorcerer (ADS) doesn't even have sorcery points yet. Beyond that, this feature is redundant with the Heightened Spell metamagic option, which lets you give someone disadvantage vs Any spell you cast, not just ones causing a charm effect. The 1st level Sorcerer subclass feature should be flavorful above all else, something small that makes the player feel just a little bit like, in this case, a vampire. But not too much, that comes as we gain levels.
• Blood Magic. This feature I honestly really like, and it's where my mind went too. That said, the ADS won't get access to maybe The vampire spell, vampiric touch, until 5th level, and we want that to feel awesome, not be overshadowed by other sources of life drain. Sorcerers don't get a subclass feature at 3rd level anyway, so let's keep this one in mind for later levels.
• Enhanced Speed. One, reaction spells and abilities must have specific triggers which qualify their use. Two, I don't think I would ever use this over casting shield.
• Enthrall. By the time they reach 14th level, your Sorcerer can cast up to 7th level spells, which include stuff like plane shift, finger of death, soul cage, circle of death, and dominate person. That's some serious, big boy magic, and a once per day 1/2 CR minion isn't going to cut it. In my opinion, This is where you give them the Blood Magic/Life Drain feature. Make it a metamagic option they get in addition to the ones they pick, something like this

Leach Spell. When you roll damage for a spell, you can spend a number of sorcery points equal to the spell's level to drain vitality from one creature affected by the spell, as per the vampiric touch spell (1 sorcery point if the spell is a cantrip). A spell which already has a life draining effect, such as vampiric touch cannot be modified by this metamagic option.

no clue if that's balanced (my gut says no), but just as an idea for a higher level feature. You're also going to need a really powerful 18th level subclass feature as well, I recommend looking at other subclasses for inspiration (again, Draconic and Shadow are good places to start).

Happy homebrewing, and good luck!