If you're not worried about the payment then what is it that bothers you about the legality of the client's business or his identity? Provided he's not selling illegal firearms, performing scams or distributing child pornography I don't see any reason to worry about it. On top of that you will see his site, so you will know what kind of business he is in from the start.
It's not like everyone is googlable or should be. I'd say being able to google somebody's name and get tons of results is the exception rather than the rule (and thank $DEITY for that).
Your concern for getting direct access to his website is valid though. If the customer is not using any VCS I'd suggest to him that he starts doing so - there are plenty of options.
If the client can provide you access to a testing/dev environment, where you can safely play with things without breaking production, I'd pick that option as well.
Sure, the big difference is that at a University they go through a bunch of things that come up and they talk about how to avoid them and why. So everyone learns how to do a bubble sort and then you learn why that is so slow, both from an analysis point of view and an algorithmic point of view, and then you move on to quicksort or insertion sorts.
Whereas someone just picking it up as they go along might independently come up with a bubble sort, it works, and then just keep using and reusing it because hey, it works doesn't it? And then when its the bottle neck and they need something better they don't even think to look at the 'working' code they look elsewhere. It is a sort of unintentional blindness.
I'm interested in machine learning, though I haven't had much exposure to that yet. I'd be interested in most things, probably.
I'm a rising sophomore, though I've self studied a few of my future classes (data structures and programming in C, discrete math) CS 223 and 202 at Yale, respectively. Apart from the languages and frameworks I listed above, I'm comfortable with HTML and CSS and ready to pick up anything new.
I live in California, though I can be anywhere if if the internship is reasonably paid.