Caster level and spell level are two different things. Cure light wounds will always1 be a 1st-level spell for you (that is, its spell level is 1st). But your caster level can and does change—it starts equal to your cleric level, and can get bonuses on top of that, like the one from the healing domain.
Meanwhile, cure light wounds says “you channel positive energy that cures 1d8 points of damage +1 point per caster level (maximum +5),” so you can see how your caster level helps—it increases your healing. At 1st level, you would normally heal 1d8+1—with the healing domain, you instead heal 1d8+2. And that keeps increasing until 5th level, when you heal 1d8+5 whether you have the healing domain or not, because of the maximum on cure light wounds (healing domain keeps improving other, higher-level healing spells, which have higher caps).
Now, none of this has anything to do with preparing spells. The ability that domains grant you—the caster level bonus in the case of healing domain—is just an ability you have for having the healing domain. It applies to all healing spells ever—that is, all spells that say Conjuration (Healing) in their description. It is not limited to the spells listed in the healing domain. It’s not even limited to cleric spells—if you were to multiclass later on and take a couple levels of bard, the healing domain bonus would apply to cure light wounds cast as a bard.
And caster level in general also has nothing to do with preparing spells or how many you get each day. Your daily spell slots are based on your class level,2 not your caster level. Caster level is also based on class level, but separate bonuses to caster level, as from the healing domain, don’t affect your spell slots, just your caster level.
Rather, your daily spell slots is just what is listed in the table for your level, plus extras for a high ability score—Wisdom in the case of clerics. Anything else that affects how many spells you can prepare, or which spells you can prepare, will have to say so explicitly.
But domains, healing domain included, do say they interact with preparing spells, via their attached spell list. The cleric class gets an extra special spell slot—the domain spell slot—at each spell level that can only be filled with the spells attached to one or the other of the domains that cleric has. Regular cleric spells cannot be put in that slot. So if you choose the healing domain, you can put cure light wounds in the 1st-level domain spell slot, because that spell is on the healing domain. If you do not choose the healing domain, you cannot choose cure light wounds for that spell slot, even though it is a cleric spell—the domain spell slot must be from a domain. On the flip side, domains can offer spells that are not cleric spells—those cannot be prepared in your regular slots, but only in the domain spell slot. So if you had the animal domain instead of healing, you could not prepare cure light wounds in the domain spell slot, even though it’s a cleric spell, but you could prepare calm animals in that slot, even though it isn’t normally a cleric spell. However, you could not put calm animals in any of your non-domain spell slots.
Please note, however, that the healing domain is very weak—as you pointed out, those spells are all cleric spells anyway, and what’s more, they’re bad cleric spells (except heal), and the bonus you get to them is very small (1 hp isn’t much even at 1st, when each attack can be about 10 damage; at higher levels, where characters can deal hundreds of damage, it’s nothing). The best way to help your allies as a cleric is to buff them so they avoid taking damage in the first place—either because you protected them, or because you allowed them to eliminate threats faster. Then use a wand of cure light wounds3 to heal up between fights (a party of four should be able to pool wealth and buy such a wand before even hitting 2nd level). In an emergency, a good cleric can spontaneously swap one of their 1st-level spells for cure light wounds, so you should never prepare cure spells. The only hp-healing spell worth preparing is heal itself, which is amazing. The rest suck.
The things you want to look for in a domain are
A good granted power
The freedom of movement effect of the travel domain is available many, many levels before anyone else can cast that spell, and even after that, freedom of movement is a fantastic spell.
The daily reroll of the luck domain can really save your bacon at the most crucial time.
Useful spells not otherwise available to clerics, for examples:
Travel domain has a number of things, like fly, dimension door, teleport, and greater teleport.
Trickery domain too, with invisibility, and very high levels, polymorph any object and time stop, some of the most powerful spells in the game.
At higher levels, protection offers a number of valuable spells—mind blank and prismatic sphere are amazing protection.
At the highest levels, the shapechange spell is one of the most powerful spells in the game, making the animal domain pretty valuable for that.
Not actually always—Heighten Spell exists, for example, and there are other ways. But anything that changes spell level is fairly unusual.
Including things that stack with that level, like certain prestige classes.
A wand of cure light wounds is the most gp-for-hp efficient core item. If you have Spell Compendium, a wand of lesser vigor is slightly better. Both can be used by any cleric, no special rules or checks necessary, since both spells are on the cleric spell list.
