I'm sorry that the paint job has not held up the way that one would expect.
If it were me, I would be having a discussion with the painters. Paint preparation is 90% of the job in my experience. I painted to put myself through school, and have always been interested in paint chemistry and coatings.
Sadly, I think that if it were me, I would be replacing your door trim at this point, and hoping that the door frame can be repaired. There are "Dry-rot" epoxies to solidify rotted wood, but if the water is getting in higher up, that won't help. Is there standing water at the door when it rains? As to the algae, I use "Wet-n-Forget". I would try that on your roof to see if it improves things. If it doesn't, I tend to reach for more potent cleaners. Was this an issue before it was painted as well? E.g. a north slope that never gets sunshine?
Take the following with a grain of salt as they say. I'm a Sherwin-Williams advocate, based on decades of experience with their paint and others. I do use other brands. We currently have Sherwin-Williams paint on the house and outbuildings and 90% off it is in good condition 19 years later. The material that isn't 100% is on surfaces where the underlying material is failing, or has failed, so not really the fault of the top layer. (Think 100 year old paint on 100+ year old wood that was repainted 19 years ago with a new layer.) Preparation is 90%...
However, Sherwin-Williams makes an enormous range of paint and paint quality. Here the low end paint is in the $35/gal price range, and the high end residential paint is in the $140/gal range. I believe that the high end paint is better. Is it 4X better? That's a tough call. I might use it on a house that I plan to be in for twenty years. Perhaps not if I only need the paint to last three years. I might use the pricier paint on areas that had access issues, e.g. multistory building, where the labor and equipment cost to get painters up in the air adds to the job cost. Commercial quality paints for special applications can be even more expensive.
However, great paint on an inadequately prepared surface is just wasting money and resources. Even a great paint needs a clean, strong, intact surface to adhere to. Sometimes, that is sanding, priming, second coat of primer, and a top coat or multiple top coats. If I am spraying, I always back roll with a roller to get the top coat into all the crevices. I think of spraying as a way to get paint onto a surface and rolling or brushing the paint as a way to get the paint into the surface. I never just spray latex paints, but that's just my personal opinion. I am more in the do it once and forget it category.
Sherwin-Williams makes a line of paint that is "self-cleaning" exterior paint called "Rain refresh" because the paint generates a cleaner from sunlight and water that dissolves off dirt sticking to the paint. ($$$, $130/gal locally) Obviously, it works better in areas that get rain periodically, and rather unusually it has a UV protecting agent that doesn't break down in sunlight. Those two items have benefits to an owner, and some geeky chemistry behind it.
All the best,
Peter
Peter has given a great summary on painting in general (and yes, I painted my way through school, too).
Water is always your enemy - let it inside next to the wood and rot (mold organisms) begins to eat away the wood substrate, releasing the paint cover. If hiring someone, the first question to ask is, "How will you prepare this job?" Painting is never as hard physically as a good prep job - that's pure hard work and few will do it right.
If the guy doesn't know what a bonding primer is...walk away. If he doesn't know how to stop rust oxidation with the proper products prior to painting...walk away. If he says he can easily repair/heal damaged interior trim (used in an exterior application) with caulk...walk away. Yes, epoxys and fiber strengthening solutions can sometimes work, but the cost, time, and skills required complicate that approach. And I've done it. You quickly are in the realm of autobody repair with those products.
I never skimp on cost for paint. As Peter notes, Sherwin Williams makes cheaper paints and excellent paints. Choose wisely. There are specialty paints as well, but that's not needed here. But a conversation with a major paint store manager is often warranted. He or she will be pleased you asked and a few pictures and some input on the job at hand goes a long way. He will also know who the good guys in town are. That's particularly important today. EPA regs (please, no politics here) mandated major changes in the fine oil based paints we used to use - hybrids are now what are being used to reduce outgassing and it took a while for the major players to get that right(er). A Benjamin Moore store will have excellent products in its pricy top tiers. Another in the East is Porter Paints, but they were bought by PPG so I guess I don't know as much as I used to (funny how that works). And though I always will want the old oil based stuff, good latex products have really impressed me lately. But, make sure a recommended bonding primer is used between any oil and latex transition...
As Peter noted, the damage to the trim and the exterior location might suggest replacement with a composite product. But read the specs carefully - wood is fairly stable in length, but composites are not. Be aware of how much they change in length due to temps and use exactly the caulk, glue, nails, and procedures recommended by the manufacturer.
I'll second the "Wet and Forget" mold/mildew treatment if needed. I use it on some damp areas of brick and stone. It works, but you will have to re-apply occasionally. Getting rid of the moisture would be better...
Painting fences...no good answer for looks. There are houses in New England that had clapboard siding that lasted for hundreds of years without painting. But, we do like the look of a white fence... So, cut the posts on a slant, don't put a vertical face plank over your joints at the post, and try to heavily dress the top of the post with paint to protect the relatively untreated core of the post. Buy posts from a supplier that will provide a certification of how much "treat" is in his posts. And every 35 years re-do some of your fence (like I had to do last year...). Amazingly, my wife did not want to help with that job like she would do in earlier times... The best white fence paint I have found (and I'm in KY) is "Town and Ranch," now a division of Farrell Calhoun. I don't spray - just feel brushes and rollers get the paint where I want it to go better. Oh, and don't paint the underside of the fence planks - just holds water and no one sees it anyway.
Best of luck