A lot of people have given some very useful advice, so I'll add to it ...
The spindle is oak, the hearth is poplar.
In general, I'd recommend harder wood for the bearing block and softer wood for the spindle & hearth. So, say Oak for the bearing block and I've had great success with Sycamore for the spindle & hearth. Also, as others mentioned, there looks like far too much friction at the hearth end - your wasting energy and heat there. It's not especially natural, but to get going, I put some soap into the hole in the hearth and now it's practically friction free ... works a treat! Spit & leaves works well for a more natural solution. The spindle & hearth should be dry too - green wood will just makes things a whole lot harder!
Is there a way to stop my bow string 'riding' up and popping off from under the bearing block?
You need to try and keep the bow horizontal, ie parallel with the ground. Personally I've found angling the bow up slightly helps stop the cord twisting together and making the spindle popout, but with most things, it's personal preference & trial & error.
Why does my cord keep snapping?
I've only had the cord snap once and that was when it'd worn through after being used about a dozen times. Sounds like either you need some better paracord, or maybe you're making it too tight, but if the spindle is spnning ok, then it sounds fine.
I get lots of vibration and squeaking and when I put more pressure down the spindle seems to bind so it only spins in one direction with the cord slipping around it without moving it on the return stroke.
The cord slipping around the spindle is usually when the cord isn't tight enough, try tightening it. Also, you may find that as the bow is used the cord naturally loosens somewhat, so I've found holding the bow & cord together works well - you can then close & open your fingers to tighten & loosen the cord on the spindle as needed, so if it starts to slip, you can make it tighter.
There was a great video on YouTube which stated that you should basically be able to do this in about 1 minute - if it's taking longer than that, then it's likely something is wrong and you should look to see what that is before continuing. This principle has really helped me nail the technique of getting this to work, so hope it'll help you too.
Sounds like you're almost there .... don't give up, it just takes some patience & soon you'll have your first fire by friction
PS Here's that video:
http://www.youtube.com/watch?v=PlxujVPiN6U