Can the spell from Aberrant Dragonmark trigger a Wild Magic Surge?
Yes. First, observe the trigger condition for Wild Magic Surge:
Once per turn, the DM can have you roll a d20 immediately after you cast a sorcerer spell of 1st level or higher.
Next, observe the language of Aberrant Dragonmark:
In addition, choose a 1st-level spell from the sorcerer spell list.
In similar fashion to this answer, the spell learned is a sorcerer spell, so it is eligible for triggering a Wild Magic Surge.
Can I use my other spell slots to cast the spell?
Yes. The Sage Advice Compendium contains a ruling about the Magic Initiate feat that applies in this case as well:
If you have spell slots, can you use them to cast the 1st-level spell you learn with the Magic Initiate feat?
Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.
Since the spell learned from Aberrant Dragonmark is a sorcerer spell, you can use your sorcerer spell slots to cast it, as confirmed by this ruling form the Sage Advice Compendium. Additionally, this answer outlines a different but applicable argument that draws the same conclusion without consulting Sage Advice.
Since you did mention pact slots, you would also be able to cast the spell using one of your pact slots. The multiclass spellcasting rules say:
Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.