Technically you could say it works, but it's not something you should expect a GM to just go along with. It's called Arcane Strike; presumably it's supposed to scale with arcane power.
Arcane Strike is pretty lousy on a Paladin anyhow, since they have more important uses for their swift actions and they probably don't need the small damage bonus much.
Nope, the CRB Faq shoots down that idea with the following:
Spell-Like Abilities, Casting, and Prerequisites: Does a creature with a spell-like ability count as being able to cast that spell for the purpose of prerequisites or requirements?
Only if the pre-requisite calls out the name of a spell explicitly. For instance, the Dimensional Agility feat (Ultimate Combat) has "ability to use the abundant step class feature or cast dimension door" as a prerequisite; a barghest has dimension door as a spell-like ability, so the barghest meets the "able to cast dimension door prerequisite for that feat. However, the barghest's dimension door would not meet requirements such as "Ability to cast 4th level spells" or "Ability to cast arcane spells".
Yes, Arcane Strike scales with your highest caster level.
No, you will not qualify for it with a trait, feat, or racial ability that grants a SLA.
If you want to take the 1 level dip into an arcane casting class. Go for it. I recommend either magus for the arcane pool or bard.
I'd throw in eldritch scrapper sorcerer for a one level dip, martial versatility being fun.
Also the feat 'caster's champion' is interesting. Only usable 3x per day but requires no dip, just an arcane caster ally.
And even if it did grant it as an arcane spell, the prerequisites FAQ above would preclude it from being used to qualify as "able to cast arcane spells".