Generally speaking I'm sure that it's possible but just to be sure.
It seems evident enough that this would work with the classic schools they are after all the same school, but how about with Elemental Schools?
They are not spells divided by spell schools, but each school gives you a list of spells that apply.
Say for Fire, can you prepare a Piercing Burning Hands from your 2nd class bonus spell slot?
Or for Void, which includes some spells that are not really in the Wizard's Spell List, such as a Keen Senses or Guidance.
Bonus question: can a void wizard prepare those spells in regular non class slots?
Given that it later goes on to add keen senses to your spell list, sure. But then again, guidance isn't. So does Void give a school slot to cantrips and you have to have guidance in it?
Add the following spells to your wizard spell list at the listed spell level:
1st—keen senses, 3rd—twilight knife*, 4th—wandering star motes, 6th—true seeing, 8th—moment of prescience.
[...]
Void Elementalist Wizard Spells
0th—guidance
1st—gravity bow, keen senses, shield, true strike
This is confusing (the first time I read it as add these spells to your spell book) because it omits Guidance and also adds some spells to your Spell List that are there anyway.
Now with domains there are more frequently spells that don't appear on the Clerics Spell List but instead give you a couple of spells that can only prepared in Domains Slots (except of course spells that are already on the clerics list).
For example the Travel Domain is full of them, such as fly and dimension door. Say you want to extend fly, you would cast it from your 4th level domain slot, no problem. Correct?
For your fire question the answer is yes.
There is no restriction that says the elemental spells can only be cast in the bonus slots. The restriction is on the bonus slots which can only contain elemental spells. In other words you may use your normal spell slots for elemental spells so that means the void wizard can cast what are normally not wizard spells.
Another thing is that when a spell is given to a class it becomes a class spell for that character.
The answer to your cleric question is also yes.
Thanks, so Guidance is on the wizard spell (of a Void school wizard) list (even though not specifically mentioned) and you know it from the start since you start the game knowing all the cantrips (except opposition school) and rare ones like scoop.