ULN2803a relay connection
The ULN2003/2803 uses a Darlington output stage and this drops about a volt when activated so, if your 5 volt relays need a minimum voltage that is greater than 4 volts you might have problems. I'd consider using logic level MOSFETs directly driven from the IO pins. You will also need flyback diodes across the relays.


  Also, am I under the correct impression, that the use of the ULN2803a
  inverts my logic?


If you connect your load (the relay) between output and Vcc (5 volts) then a logic 1 at the input will drive current through the relay coil.


  And on a side note, any suggestions on how to improve my schematics
  are welcome. I'm just starting out and I would image that my schematic
  probably isn't that well to read.


It's not bad - I've seen a lot worse. I'd consider using the GND/0 volt triangle symbol in several more places and removing the GND connections that string across your diagram. It just makes it easier to follow.


  Can I use the rail for the COM port


Yes you can but my preference would be to get rid of the 2803 and use MOSFETs and flyback diodes for reasons already mentioned.