luhcforgh: Only the little curly wire thing (bottom left) was included - is this an antenna? If yes, where/how do I connect it, as just seems to be a piece of wire? Otherwise, will I have to buy an antenna like the one shown on the right of the image, or does it work without it? Are there any antenna specifications to consider?
Both are antennas, without one SIM800 cannot connect to network. The curly wire should be soldered to the top left hole (mine is labeled NET on the bottom). If you have the other antenna, just plug in on the u-fl connector (top left round connector).
luhcforgh: I have not yet been able to connect to the network, so perhaps this is the issue. I will also try lowering the TX/RX voltage from 5V, even if it seems as if I can communicate well with the module. I will also raise the maximum current to 3A, even if I don't think I ever got close to the limit. I'll try to remember to update this post if I reach any conclusions!
RX/TX are not 5V tolerant, do not connect more than 2,8V. Read previous posts to see how to create a level shifter.