A relatively thick sidesplash/backsplash can cover that gap. I'd fill it with caulk backing first, as caulk won't bridge that gap alone, then caulk it, and then install some type of backsplash/sidesplash.

Regular grout or fill the void with caulk grout. Colors are available to to match most grouts, a simple one shot solution for a minor issue.

Baseboard will look strange here. I was the one who suggested you make a block for this area. But here is another idea that will look very good. You could use some full profile door casing. By full profile, I mean the kind of case that's used with corner blocks because it is a full 3/4" on both sides----not tapered. Some of it has flutes or beading that would look nice and some of it is 5" wide. I would use something wider than your baseboard. Here is something else for you to look at----it might give you some good ideas. This website is one I found a while back for corbels, but they also have some neat wood panels that you might find useful. It's under "new products" on their site. www.osbornewood.com

Holy close ups. You need to back up and take a full picture but for starters there appears to be no door sill. The repair is to pull the unit and install a sill and reinstall or since that looks like a run of the mill steel door buy a new prehung unit and swap them out. Since your picture is so close up it makes it almost impossible to get the full picture of what is going on this is only an educated guess at this point. And from the looks of it the jambs were not procured with the proper jamb depth and they jury rigged extension jambs from plywood instead of pine trim.