I still have the feeling that Java is being used in a lot of web development. But it's usually on the more business-oriented-no-mainly-tech-big-company kind of developments, which typically are less open then new startups that have to get some traction and promote their own work, as well as more interested in technology. So, even if it is used in a lot corporate web sites, you'll probably never know, as they won't really care to tell publicly about their technology stack.
That said, commenting all the original questions...
Is it a weakness of the language?
Compared with other languages like Python or Ruby, Java is verbose and tends to need more code to do similar stuff. But it's not just the capabilities of the language, also the community surrounding it and the kind of developers that uses those tools.
So, most of the modules and tools on Python, Ruby, PHP, etc are open source and are easier to find than in Java world, just because this one is more focused on giving (and charging) services.
For example, the Ruby community is really really oriented to web development, so every developer that is able to use Ruby will know about the problems and available tools for a web project. That is not necessarily true for Java developers, that could have been working on other kind of systems, like reporting systems. Of course, any good developer will catch up, but the perception is that the average Java developer is less worried about learning new technologies and new languages.
Is it an unfair stereotype of Java because it's been around so long (it's been unfairly associated with its older technologies, and doesn't receive recognition for its "modern" capabilities)?
Java is not really that old, and, being fair, it has greatly improved. It was the cool, relevant platform about 10 years ago. But since then, there have been new platforms with newer problems in mind, like Ruby on Rails. The core sector of Java has been mainly the corporate world, with different problems, so the people searching for new projects outside that has been looking for different tools.
Also, the main advantage of Java design, being multiplatform, is not as relevant today as it was before.
Is the negative stereotype of Java developers too strong? (Java is just no longer "cool")
That has also some truth in it. Java still is the language to learn "to get a job". So, if you don't care, but just want to learn something to earn money, you'll end learning a little Java and not caring ever again to improve.
Again, is a lot about perception and visibility. There are tons of great Java developers that are coding without sharing their knowledge, while there are lots of PHP developers, maybe not as good, that are writing blogs and collaborating into open source. That leads to think that the PHP developers are better than Java ones, as you have certain feedback about them.
Are applications written in other languages really faster to build, easier to maintain, and do they perform better?
I'd say that they are faster to build. The principles of languages like PHP, Python or Ruby makes them quite good for generate software that can change constantly. E.g. Dynamic typing makes easier to change an interface. In Java having a well defined interface is important, which leads to more stable (and difficult to change) interfaces.
This is very important in a new startup, which main problem is to get a product before you run out of money.
About performance, it is very very easy to misunderstood the needs and try to use magic tricks to achieve the required performance, like "Java is faster than Ruby. Period" or "MongoDB is web scale". Reality is more difficult than than.
Is Java only used by big companies who are too slow to adapt to a new language?
Definitively, having already an existing team of Java developers in the company, makes easier to keep using the same language for new projects. This is perceived as "the safe bet", specially if the core of the company is not technology.
But, anyway, Java is not ONLY used on big companies, there are still a lot of startups that uses Java for cool stuff (For example, FightMyMonster or Swrve uses Java extensively), but I'd say that the general tendency in the startup scene is to use other languages.
That is also a way of getting people, as most people will be more exciting to work with Ruby, Python or PHP, perceived as more "friendly" and "fun" languages than to work with Java.