Sounds like you need something like this: http://www.rutland.com/maintenance_html/105.html it's a self-adhesive strip of insulation that you use for fireplace inserts and the existing hearth. If you do use mortor, you'd need the special heat resistant "fireplace mortar" which Rutland and other companies sell.. Jay

Water will take the path of least resistance so it doesn't matter how far away the water source is located. If the soil was loose when your home was built the water may have formed a path back then. Over time it could have eaten the dirt that is under the concrete at the bottom of the steps and things are now starting to tip into a hole. Once water that is running towards your basement wall comes in contact with it, again, it will take the path of least resistance going downward by going between the dirt and the wall. Plan on having to seal the basement outside wall, replace the drain tile that should be near the bottom of the wall, and maybe even install a new staircase. A shop vac attached to 10 or 15 feet of 2" pvc pipe that has a couple 90 degree units attached at one end should help with keeping the gutters clear. Just hook it over the gutter, and vacuum away the leaves and water.

If this is really bothersome you can add a door seal (like you would find on an exterior door). The doors were hung to handle a decent carpet and a decent carpet pad. These can be as much as 3/4" - 1" above the subfloor and the doors need the clearance. If you replaced the thick carpet/pad with a thin engineered hardwood (less than 3/4"), you will find the doors to ^^this^^. A thicker under pad (like 1/4" cork) would have dealt with this quite nicely. As it is now, you either live with it or you add a door seal at the bottom. If it REALLY bugs you and you INSIST on a perfect door, you will purchase a new door. Good luck. Floors look nice.

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.