Not sure what to ask for - I've been using Inkscape for about 10 minutes, but hopefully someone will be able to help me out. And yes, I did google, but it seems like the tools in Inkscape have other names than the ones in Gimp, which I've been using until now, so the sites I read after searching made no sence to me.

This is a part of my picture:

It's something of a mess, so I need the lines to be as narrow as possible, clean, all black and with no pixel edges. How do I do that?

Use the pen tool Set the width in the Fill and Stroke dialogue box.

Do you mean drawing the line over again manually? Any way to convert(?) the lines I've got into these pretty darn nice pen tool lines?

It sounds like you need a bit of a primer into vector graphics: Inkscape and The Gimp create images in fundamentally different ways, so the sort of tools you find in a bitmap editor (such as The Gimp) either don't apply, or work differently, to those in a vector editor like Inkscape.

I suggest starting with the prequel article from my series of Inkscape tutorials: viewtopic.php?f=6&t=11981

You might also want to read the first two or three tutorials to pick up some of the basics. To convert bitmaps to vectors you can either trace the lines manually (this usually gives the best results, but requires patience), or use the Trace Bitmap dialog. Parts 16-21 of my tutorials cover both approaches.