An external monitor for my macbook late 2008. Trouble?
To answer your questions:


Per the MacBook specs, your MacBook shouldn't have a problem pushing video to a 1080p display, as the GPU can support up to 2560x1600 resolution.
You'll want to use the Moshi Mini DisplayPort to HDMI adapter if you want to use HDMI. Note: on your model this will do video only, not audio. This shouldn't make a difference on your display, but I thought you should know. Support for through Mini DisplayPort is added in the Mid 2010 models.
You can accomplish this by using clamshell mode.
As you'll read in the above clamshell mode support article, you actually have to use an external keyboard to get it to work in the first place, so you're good here. You can use a wired or wireless keyboard/mouse for this. 


Hope this helps. Good luck!