1. I always prefer wired connections over wired. Wireless can't touch a gigabit wired connection.
2. Distance can vary greatly for wired connection. Signal traveling through walls decreases the distance greatly. Is the router in the same room as the HTPC? Other end of the house? A different floor? It makes a huge difference. Things like the quality of the wireless NIC and router also make a difference.
3. I prefer Cisco/Linksys, Netgear, or Buffalo routers. Cheapo brands like Belkin, Tenda, Encore, and others, tend to be junk. Unrealiable, poor signals, and cheap feature lacking firmwares tend to plague these brands.
4. In my opinion, no. The *only* situation I can think of where a wireless HDMI adapter would be worth it is if there is *ABSOLUTELY* no way you can run a cable. For example, you need HDMI to get from a table in a conference room to a projector / TV, your TV is mounted on a wall that you can't pull cable to, or you live in an apartment where you can't run cables. Otherwise, they're extremely overpriced, and typically the range on them isn't great and they all require line-of-sight operation to function properly.