My webpage counter does not say that many people visited my webpage since Nov. Is this a bad sign? Do hiring committees really visit candidates' webpages?
I wouldn't worry about this, since hiring committees do not necessarily visit web pages, even if they plan to make a postdoc offer. If your application was reasonably detailed and your papers are available elsewhere (for example on the arXiv or as part of your application), then there's no reason why someone would need to visit your web page. Even if your papers aren't available elsewhere, the number of visits you might see is both small and unpredictable.
Are most of the good positions such as "** assistant professorship" already taken around this time?
Almost all of them will be offered to someone by the end of January, but those offers will not all be accepted (so there will be multiple rounds of offers, which may extend beyond the common deadline). I would bet that most, but not all, of the most prestigious positions have already been offered to someone, but I don't know actual statistics.
Note that early offers tend to cluster on the most popular candidates, so substantially fewer people get first-round offers than will eventually be hired.
Should I send inquiries about my application this week? Or should I wait for another? I hesitate to ask my advisor to do this, so I think I have to do this by myself.
Inquiring this week is reasonable. It's worth asking not just your favorite schools, but also schools you are less excited about. (It may signal to them that they have a shot at you after all, and increase the chances that they will make an offer. At this point, you'll presumably be happy to have any reasonable offer, and once you have one you may be able to use it to provoke other places to make offers.) Asking this week leaves a little time for schools to act before the common deadline, although you shouldn't despair even if the deadline goes past.
I now feel I need to apply for more positions to avoid the worst case
If you applied to relatively few schools, it might be a good idea to add some more even now, but don't panic. Unless you're in an unusual situation, your job search doesn't sound really problematic at this point. (I can understand that it is worrisome, but there's a big difference between not getting a first round offer from a top department and not getting a job at all. Of course I can't predict how your job search will end, but I've seen people get excellent offers after having had no signs of interest this late in January.)
Meanwhile, I highly recommend having a chat with your advisor about your job search. If your advisor is not worried, it may help you stay calm, and if you do reach the point where your advisor really starts to worry, I hope he/she will have suggestions for what the two of you can do to make sure you get a job.