1. The pet has lived with the lady until he meets his old master.
2. The pet had lived with the lady until he met his old master.
3. The pet lived with the lady until he meets his old master.
4. The pet lived with the lady until he met his old master.
(In terms of tense, which ones are right? Do you have better expressions?)
5. The abandoned dog has lived with the lady until now.
6. The abandoned dog had lived with the lady until now.
7. The abandoned dog lived with the lady until now.
(Which one is grammatical? Do you have better expressions?)
Sentence 2 is correct. The others have incorrect sequence of verb tenses.
Sentence 5 is correct. The present perfect tense (has lived) means that he began living with her in the past and continues "until now."