When you're multiclassed you determine the spells you can prepare from only that class' perspective, so you can only choose spells for your AT side as a level 9 AT and if you dip Wizard you can only learn spells liek a 1st level Wizard.

An Arcane Trickster is a spell known caster like a Sorcerer, you pick your spells and outside of level ups this doesn't change.

For Counterspelling Arcana doesn't enter into it, if it's a 3rd level or lower spell it auto succeeds, if you're trying to counter a higher level spell you need to make a check. This is a spellcasting check so if you get Counterspell from Bard, then your check is a d20+Cha and half your proficiency from Jack of all Trades. If you were a Wizard then it would be d20+Int (adding full prof if a higher level Abjurer).