The description under domains is a good resource for this
Each domain grants a number of domain powers, dependent upon the level
of the cleric, as well as a number of bonus spells. A cleric gains one
domain spell slot for each level of cleric spell she can cast, from
1st on up.
So this bit covers how you gain domain spell slots. You get 1 slot per spell level (1 at 1st, the second at 3rd, third at 5th, etc.).
Do I get access to the next domain spell each time I level up as a cleric?
No, you get access to it every time you gain spells of the next level (every 2 levels in the case of a cleric).
Does it mean that, e.g., at level two I can only cast one level 1 domain spell, no matter if it's the healing or fire domain?
Each day, a cleric can prepare one of the spells from her
two domains in that slot.
Yes, you may only use one spell from either of your domains. It doesn't matter which domain you pick, but you may only prepare as many spells as you have slots.
Casting spells outside of domain slots.
If a domain spell is not on the cleric spell
list, a cleric can prepare it only in her domain spell slot.
This further clarifies that if the domain spell is not in the cleric list (like Fireball, for the fire domain). The cleric may only cast that spell as a domain spell.
You can cast domain spells in your arcane slots with just Mystic Theurge, you don't need the other stuff. If you have a level in arcanist, you can use it just as you say. Be aware you cannot use it with your cleric spells, even cleric spells prepared in Arcanist slots, as the arcanist ability applies only to spells prepared as an arcanist (even if they are in a different spell slot type).
Best Answer
If we combine the rules for preparing spells:
( http://www.d20pfsrd.com/magic#TOC-Preparing-Divine-Spells )
With those for Domains:
Then it seems there is nothing preventing you from doing it. It says you always have the option to use a lower level spell, and domain spells are prepared just like any other.