Apologies if this gets posted multiple times, but my two previous posts seem to have disappeared after I uploaded images.
Sounds like the spam filter may have been too aggressive. That might happen if adding lots of media on a post.
The Honeywell 5800 series of sensors are compatible with 2GIG Panels out of the box. Looking at the photos, it looks like you do have at least some 5800 series sensors. The 5800PIR-RES motion detector, for example, is compatible.
The panel you are showing though is a DSC panel, not a honeywell, and the door sensor looks like a DSC one. (you would want to open the door sensor and look at the model number though, a lot of sensors use the same plastic housing)
There may be a wireless translator somewhere for the Honeywell PIR, or it might have been from a system before the DSC one and was just left there unused. It is difficult to say, but as far as compatibility goes, Honeywell 5800 series sensors would be directly compatible, DSC would not be.