|Before I get into your questions, I have one question for you. Considering how inexpensive highspeed wireless is these days, why don't you just get your own???|
What you're proposing, even though it's a public WiFi hotspot, is bordering on unethical. The equipment you'll need to use the hotsport signal will cost you $$$ and you could put that toward your own highspeed connection instead of cheating off someone elses you know.
Having said that......
We would like to get a high-gain antenna to pull in the signal, and feed it with low-loss cable to a wireless router or similar device so we can access it wirelessly. Is this possible to do?
Yes, it is.
If so, does anyone have a recommendation on what router to use and how to hook it up?
You're putting the cart before the horse. First, you need an antenna to plug into the router.....right? As for router recommendations, I like the Linksys WRT54GL as it has a Linux based firmware and there are several 3'd party firmwares you can install on them. I prefer Tomato, but I've heard others mention they liked the DDRT verions.
BTW, we are capable of hacking into a router to connect the antenna.
If this is someone elses router, that would not only be unethical, it would also be illegal. This should not be necessary in any case.
If you're talking about the wireless router you plan on buying then you don't need to "hack into it" do you.
Finally, we're all 802.11g equipped here. How much would this setup degrade our data throughput speed?
This would depend mostly on the antenna you purchased. Bear in mind that wireless is not the best media and is known for latency and lag issues. If you're planning on gaming, I wouldn't waste my time as your gaming experience would suck.