If you choose the Arcane bloodline then no, there are no bloodline powers that the Halfling favoured class option can apply to. But this doesn't mean that you don't gain any bonuses for sorcerer being your favoured class, you just fall back to the standard options. You can only chose the +1 hit point or skill rank per level.

If you wanted to use the Halfling's favoured class option then you would have to choose a different blood line, one with a power that meets the requirements, such as Celestial (Heavenly Fire power).

Using the books' rules, no, you can't.

You can only choose your archetype once, when you achieve that level (e.g. 3rd lvl Fighter). The only way to get another archetype would be leveling the same class from 1st level again, doing something similar to a multiclass Fighter/Fighter.

Although, as mentioned here, multiclassing rules in PHB specifically state

With this rule, you have the option of gaining a level in a new class whenever you advance in level, instead of gaining a level in your current class.

And in Sage Advice

Can I multiclass into the same class to pick more than one subclass?

Multiclassing is designed solely for taking different classes. For example, you can be a fighter/rogue, but not a rogue/rogue.

(Just explaining to people that might get here to question the same thing and don't know why it is against standard rules, as mentioned in the text)

Hopefully, the books aren't everything

I know the same about specific official guidelines to multiple archetypes as you - I don't know any. And yet, I've created some NPCs and let my players create some PCs with multiple archetypes, although admittedly some turned out being broken. Anyway I might help:

• Create a mixed archetype that takes out some features from one archetype and adds from the other.
• Your scenario seems motivated by the background and race of the character, so it seems you want the most flavorish characteristics. From Wild Magic, you want Wild Surge and from Draconic Bloodline you want the Draconic Resilience.
• Balancing both of them together: If you give them the Tides of Chaos from Wild Magic as well, decrease the AC given by draconic resilience and don't give them the HP bonus. Also nerf some of the wild surge effects from the table as you seem fit.
• If you give them the full Draconic Resilience, don't give them Tides of Chaos. Honestly, I think Wild Surge by itself is not a "good" feature that actually needs to be balanced if you just give it for free. For example, casting a Fireball centered on yourself usually means killing yourself and half your party. There are enough negative effects already and its trigger is the DM wants to, so you can actually just give it for free and balance it through when you trigger it.

Personally, I would make them take the Draconic Bloodline archetype and just give the Wild Surge feature for free. Balance-wise it is does not seem broken to me, although I do have less experience with Wild Magic Sorcerers than almost every other subclass.