Having a gap between a structural header and a window casing (or door) is actually a desirable thing, It means any sag in the header will not immediately affect the window, The trim on the inside should easily cover the gap. You did install flashing on the outside, right? The siding will cover that.

Your door was installed poorly. Unless the gaps at the top and bottom of the latch side(the side you showed in the pictures) is the same width. If that is true, the door frame was built improperly. The good news is if the top/bottom gaps are more narrow, the illustrated gaps can be fixed. The trim has to be removed and the frame shimmed to close the gap. The bad news is that if the gap is the same top to bottom, that door needs to be replaced.

I doubt very much that these gaps are the source of bugs...there are too many other ways they can get in. Using bug spray monthly inside your home is unwise from a health risk standpoint: that practice is potentially more injurious than anything the bugs could do. However, if you want to deal with those minor gaps, white painters caulk applied by someone with caulking experience will make them disappear. If you don't have experience caulking, enlist someone who does. It would be very easy to caulk that well and neatly if you know what you are doing, and very easy to have a mess on your hands if you don't.

The gap at the edge is not the real problem. That’s worrying about the color of the band aid over a nicked artery. That is not a job from a professional qualified tile contractor. One issue in getting recommendations from homeowners is that most homeowners don’t actually know enough to be able to judge if the job was done correctly. All they know is the guy mostly showed up on time and it isn’t too obviously a fail. They do not understand that what lies beneath their pretty and expensive tile is far more expensive, and far more important, than what it ends up looking like in it’s superficial appearance. Not meaning to insult you, but that’s true here. You are noticing the superficial low quality installation aesthetic appearance, but do not have any specifics about any of the processes or materials underneath the spot bonded tile on the out of plane walls, with no visible waterproofing on the still visible wall underneath. You’re still worried about the superficial appearance, and that’s not the real issue here. If that visible gap didn’t exit, you’d never think about the other deficiencies that are pretty apparent to an experienced eye because of that gap. (You’re actually very lucky the gap is there!) And you might go on to recommend this hack to other homeowners. Only to retract that recommendation in a couple of years time when the mold shows up. After he’s done his damage to other projects, and put you in his rear view mirror. Homeowner recommendations for skilled labor projects should always be taken with a huge dose of skepticism. Professional recommendations, only slightly more weight. An electrician doesn’t understand tile. The weight to give are industry certifying bodies, and membership in trade specific groups. Industry education for tile is trying to catch up and police itself of the unqualified. Its not the $2 a square foot Depot lot guys that are to be worried about. People know they’re getting nothing quality for that. It’s the 2K tub wall guys that are the problem. They present as knowledgeable, but aren’t. They do far more damage than the day labor guys, because people have higher expectations, but don’t have the personal knowledge to verify that the contractor can fulfill those expectations. A homeowner shouldn’t have to learn all about the technical details. But in the Wild Wild West of guys with trowels for hire, that learned knowledge is the bottom line defense of their castles against Trojan Horse contractor’s who surface appear to be qualified, but aren’t. To mix a half dozen metaphors. That work is an obvious fail. It needs to be completely redone, from the framing out, with someone who understands and follows industry standards. While cost is no guarantee of quality work, it is an overall indication of the time and quality of materials being used on the job, which does correlate to the quality of the work. I‘m not talking the actual tile. That’s completely irrelevant to what goes on under the tile. The required materials and time that it takes to do a shower properly has the industry prevailing costs be in the 7-9K range. A tub surround, 4-7K. When interviewing for your new contractor, after taking the proper steps to fire this one, please use the questions from the CTEF website about how to find a quality tile installer. Then use your now educated sad experience to educate others.