I just answered a question on screen recording, went on to browse the rest of this site and found this question. Though I guess it's not meant for video editing (I'd prefer Sony Vegas, but that's not free), ActivePresenter should also be able to fulfill most of the requirements.
First up, though: how to get started. Normally in ActivePresenter, you get video into a project by screen recording, but you can apparently also just insert a blank slide into a blank project, and just drag a video file in there. Tested it and works just fine.
On to the requirements:
trimming video clips, playing one video clip after another
ability to overlay video as well would also be nice
previewing the video before exporting it
Here's a screenshot of the basics:
There's the typical "Play" and "Stop" buttons, hitting those will preview the end result.
The arrows point towards Cut-, Copy-, Delete-, and Crop To Range options.
Dragging a second video file into the project adds a new layer with that movie. You can cut/paste ranges as you please, as well as show videos side by side, picture in picture (by resizing one of them), etc.
ability to insert images or audio
You can add audio, video and images by just dragging them into the project (or inserting them from the
Annotation menu). On the timeline you can use the trimming tools mentioned above to determine when they are displayed. In addition, you can place and resize them to your liking.
Here's a screenshot with a main video, inserted picture of a pony, and inserted second movie clip as picture-in-picture:
There's a free version (which I used) that has abovementioned features, as well as a standard and professional edition (which I haven't tried).
and relatively straightforward to use (less important)
This is somewhat subjective of course, but I found this tool quite easy to work with. First 10 min screencast cost me about 2 hours of work, but after that things start to speed up. Disclaimer: I've used Sony Vegas quite intensively so I was somewhat used to video editing concepts (which helps a lot with the learning curve, probably).
This is the only one I'm not sure on. It's probably quite limited compared to Sony Vegas, Adobe Premiere, etc.
For exporting you're stuck with FLV, AVI, WMV, MPEG4, WebM (docs).
For importing clips I couldn't find any documentation, but the "Select a video file..." dialog says:
Video Files (*.mp4;*.mkv;*.wmv;*.flv;*.avi;*.webm)
I guess you'll have to have a go at it to see if your particular clips are supported.
As a bottom line, to repeat my earlier remark: ActivePresenter seems to fit (most of) your requirements, but isn't really meant as a video editing tool. Even so, it might just work for your case.