I don't think you're giving yourself nearly enough room for routing traces in that board. Is there a reason you can't make the board bigger and spread things out? That will make it a lot easier, especially for your first board. You can always make things more closely packed later on if you want to.
Trace size will be dependent on where you're manufacturing the board. Many fabs these days have 6 mil minimum trace width and 6 mil spacing but check the design rules for the pcb house you're using. That's a minimum; I would use bigger traces unless you absolutely need tiny traces. I tend to use 8mil traces as default. Use bigger traces if you need lower resistance. Use a pcb calculator to calculate trace resistance for a given width/length. Combined with the expected current, the trace resistance will give you voltage drop and temperature rise. Kicad has a basic multifunction pcb calculator built in but there are others out there, some with more features.
The NC flag in Kicad just means that you are not intending to connect anything to that pin. Normally the electrical rules checker will complain if any pins aren't connected to a net; the NC flag tells ERC that it isn't a mistake and that pin is intentionally disconnected. It does not tell kicad that you want that part excluded from the layout. If you don't want a part on your board, either remove the part from the schematic entirely, or edit its reference designator to start with a # (for example, if you didn't want R101 to be included in layout, call it #R101 instead). In this case I would recommend just deleting the part -- if you don't want it on the board, why is it in your schematic? You could also leave it in your schematic & layout, but not solder the part. In this case I would add a "DNP" (do not populate) note in the properties for the schematic symbol for the part to remind yourself or anyone else that the part should not be actually assembled onto the board.