All the curcurbits (cucumbers; summer & winter squashes; pumpkins, etc.) very easily cross pollinate as you have discovered - and the resulting seeds will produce strange looking fruit in the following year.  It's edible - but may not taste very good (if the cross was a 'winner' it would be a new, marketed variety!!!).  I can't tell you exactly what you've got, but it definitely sounds like a cross... like I said, you can try to eat it with no harm - up to you! That's why it's recommended to only buy curcurbit seeds from a very reliable seed company, where they carefully guard against cross-pollination.

In answer to your second question, I leave winter squash on the vine as long as the vine is thriving; once the vine starts to die off, I remove the squashes and leave them in a sunny spot for a few weeks to harden off (some people simply leave them in the field).  Once dried out and hardened, they can be eaten at any time.