After the new (now correct) boards from OSH Park arrived I could start trying to solder the parts. 0603 passives are no problem at all, but the LGA-16 is. It's really tiny, and the pads on my board seem to be too wide, resulting in shorts between two pins on two boards:

• First board, the two pins to the left are shorted:
• second board, pins 2 and 3 from the left are shorted:

I did my best to apply only little solder paste, equal amounts to each pad, using a kapton tape stencil:

So I asked for advice on electronics stackexchange, and the most important hint was to use a way of reflowing that allows me to poke the chip while the solder is liquid. Having no hot air gun I decided to use a hot plate. I've added my solution as an answer there, including all the details.

This time everything worked and I have a board without shorts and all pins seem to be connected:

Now I just need to find some time to write test code!