Well I 'had' an HP wireless printer, the fairly new C309G model. I got this printer on Black Friday and it gave me fits. It was an N speed printer, and all I wanted to do was set it up to my router and then loaded everything needed on my W7 machine. I could NOT get it working satisfactorily at first. I got on line with HP and via phone. All I got out of that was I had back level drivers and applications, and a location to get the latest. I did that, and after over 6 hours on the phone over 3 different calls, I was able to figure out how to make it work.
First thing though, HP insisted on adding a string of ports to Windows Firewall, even though it wasn't being used and I have McAfee's active. As the tech was adding them I wrote them down (I no longer have that info, sorry) and saw later McAfee already had those open. So, one suggestion is just DISABLE your FW as a test to see if it is causing problems.
Oh, my problems, as I remember them as the printer was returned in a week, very similar to yours. Could NOT start a scan, always got an error back the device either was not there or was busy. Printing a picture, either printed partially or none at all. Printing a page, hit or miss, usually nothing printed. Only reliable thing that printed was the test page from the printer control app?
Like I said, I did GET it working. What I discovered was the printer had an HTML browser interface, that always worked. One of the settings, USE G if there was an error. My router was switched from N ONLY to MIXED and lo and behold, the printer worked fine. However my LAN speed was cut in 1/2. Worse, there was NO WAY to restore N operations... even calls to HP couldn't get that to work. So back it went.
Maybe this isn't exactly what you are seeing, but there are some things to try. One would be call HP, I think it is free. Second it check their website and get the latest drivers and utilities. If wirelessly connected to a router, change channels or connection speeds.