does a video in this HDMI out mode count as just one click?Effectively zero clicks:
Aside from rotary shutter options available on a few high-end (>$50,000) digital cinema camera models, video exposure time is controlled by sensor electronics; all other mechanical shutter mechanisms I'm aware of remain open when capturing video (for any combination of recording, HDMI out, in-camera LCD live view, or electronic viewfinder display).
Pedantically, I suppose you can count one click for a normally closed (DSLR) shutter and zero clicks for a normally open (mirrorless) shutter, though I have no idea if DSLR firmware actually bothers to do this.
I'd also guess that simply opening and eventually closing the shutter causes less wear and tear than a typical <<1 second still photo exposure.
Also can you completely avoid using a battery with a power line adapter?
Generally, yes; in cases where battery-free DC input is not otherwise available, inexpensive "dummy battery" adapters are typically available that pass power through the camera's battery contacts from an external source.
Check adapter compatibility before buying a camera, however; make sure that a compatible dummy adapter (a) exists, (b) doesn't physically interfere with camera operation (buttons, tripod mount, etc.) and (c) attaches without a significant risk of damage to the camera (e.g., if you plan to use the camera in anything other than a totally stationary position, you probably want to ensure the battery compartment door can be closed, removed, or otherwise secured when the dummy battery is connected).