I'm a postdoc and know a lot of other postdocs, but don't claim any other expertise here. Hopefully others will weigh in as well. For what it's worth:
1. There are two main styles of postdoc, (1) continuing similar research from grad school, but in a new lab, and potentially with some new research tools/approaches, or (2) using the postdoc to get interdisciplinary training in another, related area to your own. A strong previous research record, including strong publication record and a skill set that is needed in the lab will make you competitive in both cases. I have a type 2 postdoc, which has worked out great for me— I got the cross-training I wanted, and the lab got expertise in an area they wanted to move into but that the PI doesn't have background in. Note here though that I was desirable for this position because of the skill and knowledge set I had that they wanted— few PIs will be willing to train someone wanting to switch to a new area unless they are getting something in return. The possible exception is postdocs supported by training grants/fellowships (e.g., T32 or F32).
2. Many postdocs I know, including myself, found postdocs through networking— I had another faculty member introduce me to my postdoc PI while still in grad school, then applied for and got a fellowship to work with him. Others got their postdocs by applying directly to training grant (T32) programs (though it is good to contact PIs in advance for that too), or applying to posted ads. Certainly if you are cold contacting people without ads you would want to do so in the context of suggesting writing an F32 to fund yourself.
3. I have no idea how common this is. I think like any job you need to ask good questions during the interview if you get one— e.g., get a good description of what you would be working on, expectations regarding authorship and publishing, opportunities to get grant writing experience and hopefully be co-I on grant applications, opportunities to gain experience mentoring more junior lab members. These are the types of experiences that I think have been most helpful for me on the job market and as I'm moving to a TT position this coming fall. I think most PIs really want the best for their trainees and will talk about career development and your goals as part of the interview— if they don't that might be a red flag.
Best of luck!