You describe a mixed collection of tasks, and may do best with multiple contractors.
Someone who advertises as a general contractor probably won't be interested in many of these tasks, and/or would be expensive because they would farm out the small jobs to many subcontractors, adding cost and profit on top of the actual work.
There are some handyman services that could probably handle everything. They are big organizations with a variety of specialties. They're licensed and experienced in the different trades. Many are basically general contractors, but they focus on small jobs and use their own crews.
I've also worked with individual handymen who had broad experience in construction, were well-skilled in virtually every trade, and just wanted to be in business for themselves, doing smaller, multi-disciplinary jobs start to end.
Just make sure that whoever you select has experienced people doing each task. There's a difference between knowing how to do something and doing it day-in and day-out.
Replacing an awning with a retractable shade could be handled by a handyman with some experience in construction.
Adding window trim over stucco is something I would want done by someone with some carpentry and stucco experience. "Stucco" can be done in a variety of ways, and it serves as a skin. It's easy to damage, create water leaks, etc., so you want someone with some construction experience who understands stucco.
Replacing a fence gate is a task for a handyman with some carpentry or fencing experience. This is a relatively small job, so it will be much less expensive to have it handled by someone who is already on-site than to bring in a fencing company just for that.
Exterior painting is a job for professional painters, especially if the house is multi-story, stucco, or has a mix of exterior surfaces and finishes. A handyman service may have a crew of professional painters. This is a big enough job to contract separately with painters if you work with a handyman service that doesn't regularly paint entire house exteriors.
That said, if the house is stucco, paint may not be what it needs unless your objective is to change the color, or there is extensive cracking you need to repair. The color is blended into the stucco mixture, not a surface treatment. If it just looks "weathered", it may only need a cleaning, which could be handled with low-pressure power washing.
Power washing the brick depends on what you need to accomplish. If the brick has chemical or oil stains, paint spots, lichen, or other heavy-duty staining, or covers a very large area, it may need the cleaning power of professional equipment. It it is just normal environmental "dirt" and not a huge area, consumer-grade equipment will be adequate.
If professional grade equipment is needed, a handyman service may have it. Just make sure they have some experience cleaning brick with it. Overdoing it can blast away chunks of brick. There are also lots of power washing services. They will have professional-grade equipment and some practice, and will be used to doing it efficiently.
If the bricks are just dirty and not a huge area, you might even think about doing this yourself. Professional grade equipment isn't required for simple cleaning, although it would be faster. For what you would spend delegating the task to a pro, you could buy a cheap electric pressure washer, do it yourself, and then have a pressure washer for future cleaning needs.
When you talk to contractors, they will all say that they do all of your tasks all the time. Recognize that there may be some level of BS in their claims. Push for details, ask to see pictures of similar jobs they've done, and check out references if practical. Start with recommendations from neighbors and colleagues, and consider using a service that does some level of initial screening like Home Advisor or Angie's List.