I seem to be having a similar problem. Same card. Managed mode works for me but overall card performance is just... not worth the money. Literally, iwl3945 in my notebook seems to have a better range. Just as an example, after "iwlist dev scan":
iwl3945: about 8 networks
alfa + 12dB omni: only my ap that is 3 meters away
Same results with airodump-ng.
Seriously, sometimes it looses the signal from my ap if i go the other room. I've tried both rtl8187 and r8187. Im out of ideas. Just one question (also to other awus036h owners). Are you using that double usb Y cable? I got just a normal usb -> mini usb and am hoping maybe thats the problem i.e. not enough power. Could it be?