Look at Schluter edging - try the Jolly profile. It needs to be installed at the time the tile is being installed to create a clean edge. Also, the cuts don't look great. Your tilesetter needs to ensure a proper sharp blade for cutting the tile. Looks like glass and maybe marble in that tile so they need to be sure they have the right tools to work with those materials.

when you are all done installing tile, leave the WALL edge un grouted. Grout everything else and come back when the grout is dry and buffed.

Use blue painters tape and tape over tiles, and along the wood wall trim edge.

Caulk neatly, a "grout line" with a caulk that matches your grout color. Remove the tape. Tada!

maybe try a spacer. the tile stores usually carry them or home depot. you either can use them as spacers or there are L shape one for edges.

"I have a mosaic shower border being installed"

This tells me you are paying someone. First thing to do is have them remove all the mosaics. Seeing the material is glass they would need to be set in a smooth bed of mortar. Basic glass installation 101. All those ridges you see will show through the glass. As for the jagged edge. That's just lack of knowledge. Glass could have been cut with a good glass blade or cut / scored from the back side. No caulking or grout will hide the jagged back from being visible.

Are all the mosaics flush or slightly proud of the wall tiles? They should never be recessed!

With the mortar mess it is quite clear you do not have a Tile Setter on your project. Any in progress photos would be helpful.