Glad you've got started with them so quickly! I just tested this on Mac and I have the same problem. I've checked the Web IDE's code, and it appears to be chrome that isn't reporting the ports correctly. You can solve it by going into a terminal and typing the following:
sudo ln -s /dev/tty.HC-05-DevB /dev/ttyusbmodem12345
I've just contacted someone who was working on serial support in Chrome, so hopefully I'll get a better idea what's going on soon.
The bluetooth disconnect is a pain. I'm actually not sure if it's the computer or the module itself - I'll have to try and see if it happens on a PC (how long did you leave it?). If it's a module problem it's probably solvable with it's AT-command interface mode.