Go wired or get a better USB adapter. I have one of those Netgears and it's rubbish compared to the
Rosewill unit I replaced it with for the old lady's PC. Now she's constantly experiencing peak downloads from the Internet, the same speeds I reach via CAT5e a few feet from my router. Her internal network speeds are about 1/5 or less that of my 1Gbps Ethernet connection (peak transfers I've seen around 115MBs or so), but a big part of that is the USB 2.0 limitiations (480Mbps or about 20MB/s) and having a USB adapter...it's still the fastest USB adapter I've ever used. I've owned several, a couple costing way more than I'd care to admit. The cheap Rosewill smoked them all and has impressed me greatly with USB wifi adapters from this company. The antenna helps keep a solid connection, and the drivers it uses are pretty damn good. Wiring is the best route if you want it all though, but there are affordable and decent USB wifi adapters out there should you want to continue down that road. Sell the Netgear one though!
I don't know about needing to go with CAT6 really...you could and it would be great for the future, but you don't need to by any means. Especially if you don't plan to upgrade to anything that can take advantage of that cable's bandwidth ratings in the future on both the NIC and switching/routing side.
CAT5e will do just fine and is still a good margin cheaper at least in my area. Especially if you aren't terminating your own or buying bulk. CAT5e does gigabit speeds just fine, CAT6 is a better cable for sure, and is 10Gbps ready...but really when CAT5e can be had so cheap and 99% of folks' home routers only support 100Mbps connections, it's tough to recommend imho.
Good luck and keep us posted with what you decide to do!