There's nothing wrong with full hopping or short hopping a Zair, it just depends on your situation. If you need to release a Zair quickly, then short hop it. If maybe you need to jump over something like Samus' Charge Beam and then Zair when you finish jumping over it, you might want to full hop. You get me? No discrimination between the two, it's always situational. Short hopping is most useful though, since it's quicker, so you might want to use that more often.
Good follow-ups after Zair:
Zair -> Nair
Zair -> Grab
Zair -> FTilt (depends on spacing)
Zair -> Dash Attack (depends on spacing)
Zair -> USmash (depends on spacing)
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If you throw a bomb below you while falling and it hits an enemy, Dair is not a bad follow-up as long as you're sure you can land the Dair. Again, it's situational. When you throw a bomb down and it hits, you have the option to land pretty much any aerial you want, it's just a matter of choosing the one that best fits the situation. I would use Dair if the enemy is at lower percents, that way, I can try to rack damage on him by chasing him with a Dair pogo -> Zair -> Follow-up.