Answer 1: Your logic would suit recording on analog tape. As answered, this is digital and it is about separate and individual clips. However, the TM900 comes with software called HD Writer. It does a few things not found in most NLEs (editing software programs). One of those is to glue the clips you select into a single new larger clip (or file). It does this while perfectly preserving the original quality and without altering the original clips.
Think of the record button on both the TM900 and FZ150 as a "start/stop". They both work the same way.
Answer 2: You can review any clip at any time that you are not recording. On the TM900 there is a selector knob that puts it in playback mode which presents "thumbnails" on the touchscreen. On the FZ150, I'm not sure yet. My wife has had it only a week and I'm afraid to touch it, so have not tried. Their is a "Play" button on the back and another thumb pad for selecting the clips.
Answer 3: The TM900 has a mini HDMI out port. Plug the other end into a TV with HDMI in and the TV screen is a duplicate of the camera screen during recording and playback. If you use the camera as a media player, the only way to have a TV screen completely clear of labels and icons is with the "display" button on the remote. One of the nice features of HD Writer is that the capabilities are there to move selected video back to the camera so that you can use it as a media player on ANY TV with the right cords.
The FZ150 has an HDMI out port. But, according to the manual, the camera screen goes blank when you plug into a TV. I haven't tried it yet, so I can't confirm it.
I can confirm the TM900 will suit your "kids playing sports" purpose. I don't own one, but I own it's close cousin the SDT-750. I have granddaughters playing fast pitch and it records superb video. If you take video in 1080p you get a limited, but useful version of "High Speed Video". HD Writer has a function where you can capture individual frames as .jpg photos. You can click frame by frame through the 60 it takes each second. I have an ink jet printed 8.5 by 11 on my office wall with my cute and charming granddaughter in full "slugger" stance with the slightly blurred ball inches from her bat. I would never have that photo if I was shooting with a camera. HD Writer doesn't turn your clips into slow motion, but my favorite NLE, Adobe Premier Elements 10, does it very well.
If you goal is video and the price is not too important, I think I would push you to the TM900. I don't think anything does better in daylight situations like kids playing sports. The FZ150 will save you some cash and take better still photos and, according to reviews, you'll still get excellent video. I bought my wife an FZ150 for her birthday because of the zoom and photos. She could care less about video. I'm getting her the add on zoom lens so she can take bird pictures. With the right bird, she may actually end up pushing the video button! I will have to wait patiently before I can test the video on "her" camera.