You can only cast them once per long rest
As you quoted above, you can only cast each spell once per long rest. A cantrip is still a spell (just at 0 level, which would be the lowest level you cast it as) and therefore falls under the same requirement.
Counter-examples
Dragonmark differs in its language and requirements from things like the Magic Initiate Feat or The cantrips gained from a certain race (like High-elf.) Those simply add that cantrip to your current list and do not have a limitation on uses/day.
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.
Best Answer
It's up to the DM to rule on whether or not it counts as a Warlock spell.
It all comes down to one question: does the spell count as a Warlock spell?
If it counts as a warlock spell, then you can cast it with Pact Magic slots, since Aberrant Dragonmark says:
Pact Magic slots can be used to cast Warlock spells that you know. Since you know the spell, if it counts as a warlock spell, you can use Pact Magic. If it doesn't count as a Warlock spell, you can't.
This question addresses this problem: What makes a spell being cast considered to be a {class} spell?
Notably, there are conflicting answers. This breif answer alleges that all we need is for the spell to appear on the Warlock spell list. However, this answer (section 4) alleges that because the Aberrant Dragonmark feat specifies the Sorcerer list, it counts as a Sorcerer spell.
This is not at all clear, so it is up to the DM to make a ruling and stick with it.
If we're counting it as a Warlock spell, use Charisma when casting via Pact slots.
If it counts as a Warlock spell, and we're casting it as a Warlock spell using Pact Magic slots, follow the rules for Pact Magic: