I sent the $2000 cleanup bill to the tenant for negligence and she is refusing to pay claiming that my homeowners insurance should cover "old pipes." 

If so, is the increase a lot or is it negligible?

My insurance wouldn't pay anything because they said I wasn't negligent.

As a result, should avoid purchasing properties that require more than a negligible amount of work?

Keep in mind that the initial investment is often $0 when leasing so returns are infinite, it’s akin to getting a property with 100% financing and negligible mortgage payments.

I hate that he has put us in this position, due to his negligence.

The amount of PMI you will have to pay is negligible when you look at it from a point of view of what else that extra 16.5% of downpayment could be doing for you.

Then he discovered the judge from Housing court had previously put a demo order on the house due to a past owners negligence and not correcting numerous code violations, also recorded on title.

One day while in retirement they were all out fishing but waiting for them at the shore were police with a warrant for their arrest for negligence, do not become on of these kinds of flippers.

I can give you many examples of where i've had to hire a second contractor to fix the first contractors 'mistakes/negligence' and in the end it costed much more.