I concur with krzywica, thou you might not have to replace parts in that laptop, if you unable to identify it's components - shows possible lack of tech knowledge on your part to finish this project exactly as you imagine. I understand instead of building new HTPC you'd like to convert your older laptop?
You can achieve relatevely easy dvd library and media client with
10ft interface
As far as horsepower goes, Precision M4300 has more than enough and provided video card was upgraded it might even support video card accelerated playback, thou it sounds like it might not be required in your case.
Problem #1 - 300 Dvds in original form will take atleast 1.5Tb of space, which I doubt your laptop has that space, most likely you'll want to have enternal NAS for storage all this movies.
For media frontend, the laptop should be fine. Just install
Boxee on it and you'll be set. For remote control I prefer to use single universal remote, so a simple $50 usb MS-MPC IR receiver (off ebay) should take care of remote control part.
Problem #2: PC based DVR - while it possible to do, it very much depends on how do receive your tv now (roof antenna or bunny ears, cable/sat/fios or uverse). This WILL require high level of technical knowledge and expertise to achieve.