I'd like to just expand on a couple of suggestions since I do this a lot.
Siege is a great sanity checker and it has a real low barrier to entry. It's a standard package in Ubuntu and others and only requires command line parameters to run. So you can slam your website as hard as you want (or dare) to see where the breaking point. My experience is that few web servers degrade in a linear fashion. As load increases they get a little bit slower, then a little bit slower, then BAM your load time jumps to 20 seconds. Knowing when that happens gives you an idea of how much to worry and if things need to be done. If you don't hit the wall until 500+ concurrent users, then you have some leeway.
But note that siege is really more about testing your web server than your web SITE.
As the same poster previously mentioned Selenium is a great tool for that, however, expect to invest some time. You can actually pop off a few tests just using the Selenium IDE and running them either in your browser under Selenium RC as an HTTP suite without getting into scripting (although you will want to do that). Both the up and downside of this approach is that this is powering a real browser, so running the test on just one machine will not get you real-world world results because your machine can just not keep up. You need to be running and setting up SRC and Selenium Grid can be a daunting task, but totally worth it if this is something you will be doing again and again, especially if you have a little VM knowledge.
The guys who created Selenium now offer a commercial cloud-based RC that runs all the standard browsers and captures video of the session (which you can turn off for a performance test). It's at http://saucelabs.com and for either load testing, or care free quick tests on all browsers it's great. (I still run functional tests locally). Again, it costs, but it's reasonable.
I spell this out more because your first point is testing your site "vigorously", and I am assuming that means testing all functionality, not just testing it with a lot of energy.
I don't know any magic bullet for this except for writing thorough tests (which is where you usually get into scripting since a lot of things can re-used, e.g. filling out a form).
Last thing is, before you launch it, if you don't have a usability program in place, let some non-programmers/web-people use it and watch them and ask them to do various "tasks" (sign up, find something, no "click on"'s) This sounds like usability testing but it's too late for that, but you will be amazed how quickly people who don't think like you find bugs in your code. This should hopefully open up new horizons of areas you haven't tested.
I haven't found a "coverage" tool out there for web apps. If there is one, but I would love to hear about it.