Interesting case. I can't see the apartment complex being liable for bedbugs, based simply on the nature of them (they're brought in). The complex seems to have demonstrated reasonable care by exterminating four or five times, so they don't seem be acting negligently.

I can see their attempt to find you liable for damages resulting from your tenancy, if they can prove that it was you who caused the damage (i.e. the clean-up). That's their burden. Their proof is the existence of the bedbugs within your unit so that's pretty strong. If they can present the evidence that these creatures do not travel from room to room, your case is weakened.

The upstairs neighbor asking you to lie is pretty damning, suggesting that they introduced the pest to their unit, but that's only hearsay unless you can get them to testify that they're guilty and that's not likely to happen. Then you have the hurdle of your contract, which likely includes a pretty open-ended description of what constitutes "damage".

You would need to prove that the bedbugs did not come from you. I don't know if you can do that. You can build a case based on all conditions of the unit, and any negatives, such as the leaks, the lock, in an attempt to discredit them, but I would have to say it's a very difficult case to win on the merits. What you would need to do is win on the sympathy of the court, which is entirely possible.

You need to protect your credit. If this were a slam dunk, I'd say let them sue and dispute the mark after judgement. Conservatively speaking, your case is not a slam dunk, so you may want to bite the bullet and pay.