Yes ... new stucco, or patches on old stucco, is very alkaline and needs to completely cure and for the surface to get washed off a few times. It's often painted too early to save the builder time and mponey, which is why you see newish houses with peeling paint.

Judging by your picture,(tell me if im wrong), it looks as if that application is the scratch/ brown coat with no stucco top coat,(typically 1/4" thick). After the lathing/underlayment, a cementious based scratch coat is follwed by a brown coat of the same material that is brought out to a level to match up to the existing. The scratch/brown coat will crack normally and is important for it to sit for at least 3-4 weeks in most cases before the stucco color coat is applied. This allows the cracking, again normal, to happen as much as it can and wont follow through to the color coat. You can caulk with an elasomeric caulking after "v-grooving" out the cracks, but i would let it set for the 3-4 weeks and top coat with stucco mix top coating, then paint to match. Cracking can also occur due to a "too dry" mix drying out too fast,(hot weather), as well as improper install of the lath wire. If you go the top coat route, while waiting for the scratch/brown coat to "cure and crack", periodically a.m. & p.m wet the area down with water, Give it a good soaking with a garden hose.

It's going to depend on the door. You are going to want to cut the stucco back enough each side and top if the door has a nailing flange to fasten into the trimmer/jack studs as well as on the header. The void can be filled with foam,(low expanding foam for doors and windows), befoe your trim is installed. The trim itself should be back caulked as well as topically caulked after it is installed. I would prime the trim all 4 sides before installing. If the door has no flange, you can cut the stucco tight, install the door, foam the gap, then the primered, back cualked, topical caulked trim. I dont normally recommend relying on caulk to keep water out, but seems like the only re-course without disturbing the stucco and trimming over. The bulletproof way would be to knock the stucco back wide enough to run a head flashing over stucco mould that would trim out the doors exterior. The flashing would run over the moulding. Then self adhering bitumen type flashing installs up the sides with a piece then installed over the header flaashing. Then mesh wire/weep screed at the base followed by your stucco patching.Your scratch coat can be followed by the brown coat after 24 hrs and then let set a couple of weeks before the color coat. Your brown coat will be leveled with the existing wall surface. Grind enough of the existing top coat around the perimeter to be able to feather the new top coat into the existing. Depending on wether you have a textured top coat existing, or fairly smooth, i typically have exterior acrylic latex paint matched up to the color coat and paint the new color coat area as it is usually hard to match up existing color coats that have been on the home even if you knew the product brand that was used. That way you dont have to mess with trying to match up the color coat old vs. new. You would want an exterior flat paint.

It depends on the type of stucco and the method used to install it. I'm serious when I say look up stucco on Wikipedia. There is good information for you as far as stucco development. Some requires painting, some doesn't. It depends upon the mix and materials. If you have to paint it use elastomeric paint. It will expand and contract with the change of temperature. It's more expensive, of course! All stucco cracks. It's part of the material. The consequence would be that it could pull away. Remember, your stucco guy should do nothing but stucco and stucco painting. He's your expert. Just make him guarantee his work. Your GC, if you have one, will only have a superficial knowledge of stucco.