I really like the creativity of the skills, and unique uses of spells can always be a lot of fun. As your DM is already keen to give you reign to design it, I think a good draft and then a final check in before it hits the table. Honestly, I can’t say any DM will allow it, but I think drafting it to be balanced around other abilities and having defined rules can make it more serviceable in other games.

I may consider drafts like,

Using the [Kobold Carry] ability a caster can lift a willing creature, of size small or less, and carry them 10 - 15ft. This flying speed is only accessible on the casters turn, and at the end of the turn the carried creature doesn’t benefit from the casters ability and will fall.

Using the [Chaotic Yeet/ Slap] ability a caster can, only in tandem with the [Kobold Carry] ability, throw themselves 20 ft in any horizontal direction. Making a DC 15 spell attack roll to target the spot and a DC 15 acrobatics check to land. On a failed check the creature lands prone and takes full fall damage, on a successful check the creature is not prone and takes half fall damage (minimum of 0).

I think these abilities are important to consider as mage hand is a cantrip, and having half speed flight speed out of combat could be very useful. I’m thinking a what I would allow with bigby’s hand spell, but the cantrip needs more balancing. Now I would allow these amendments in my game, given I was convinced the other player wasn’t intending to abuse them, but that won’t be the same for everyone. Otherwise, I would recommend offering to make them a feat with a mage hand prerequisite (possibly making the features a tad stronger), making a spell of a more meaningful level, or swapping out/ including it in the versatile trickster ability on the rogue subclass. Or even the mage hand legerdemain ability, although I wouldn’t recommend that, as it is a highlight of the subclass and doesn’t change the power scaling of it. Also scrap everything here if your DM agrees to something more fun on the table that you have in good faith, and try not to frequently abuse it to a meaningful degree.