What are stud rights for the puppies in an unplanne d pregnancy?

"An oral agreement was made between the owners..." Making this an agreements/contracts issue - moving accordingly. To the extent you can...

Stop trespassing on her land, stop treating her land as your dog's personal relief station, and you will stop having this problem.

Eviction is a landlord/tenant matter - moving accordingly, so you get specific answers.

Is this a VALID CASE?

The pup may also have parvo; common in puppy mill dogs. If TX has a "puppy lemon law," review the facts with a TX attorney.

In Texas, if my dog wandered off n gets lost is it considered abandonment?

File a replevin action against whomever now has your dog.

Moving to contracts, where this belongs. As we have not seen the contract, its enforceablility cannot be determined. It is unlikely the seller...

Stop broadcasting your felonious act and retain a qualified defense attorney.

Stop wasting time filing police reports and sue on the contract. Given the sum involved, retaining counsel would be prudent.

If the attack was in your yard, and your dog was not the aggressor, sue the neighbor for all costs related to the attack.

Someone is adopting my cat and the cat is acting up at their house and pee on their property, can they sue me over that?

The cat is clearly stressed over the situation. Perhaps these people should not have pets.