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.
Yes. In 5e, prepared spells remain available as long as they're prepared.
From the D&D Basic Rules (e.g. from the cleric's or the wizard's Spellcasting feature):
Casting the spell doesn't remove it from your list of prepared spells.
Basically, 5e Wizards (and Clerics) are 3.5e Sorcerers who can swap out their "known spells" based on their spellbook (or godly mandate).
I can't speak to the exact reason behind the design decision, but I assume they wanted to enable a level of flexibility for both classes that was often lacking in previous editions.
In general, if a spell targets allies, it just succeeds if they're willing.
In the case where some mechanical element would be interfering with the spell (anti-magic or counterspell), I don't think there's an explicit mechanical way to be notified if a spell succeeds or fails.
There are no saving throws against Bless. Given that it's an entirely beneficial effect, I don't know why you would.