A pig is any of the animals in the genus Sus, within the even-toed ungulate family Suidae. Pigs include the domestic pig and its ancestor, the common Eurasian wild boar (Sus scrofa), along with other species. Related creatures outside the genus include the peccary, the babirusa, and the warthog.
It is very good to save or help any animal including dog and pig. We have a Hadeeth about a sinner who was entitled to go to hellfire but was saved from hellfire just because he gave water to a thirsty dog. There is also a Hadeeth about a Muslim woman wo will go to hellfire just because of a cat which was locked up by her and left with out food and water. It is always very good to help, feed, give water to any animal.
Touching wet dog or wet pig will make your hand or body Najis, and that will never harm you as it can be purified with water.
We are meant to be the guardians of the earth, showing mercy, especially to innocent animals. We have a duty to care for them. Therefore, even if you became najis after rescuing the animal, this is of lesser importance than showing mercy and humanity towards a sentient being that is suffering. You can easily become tahir again by purifying yourself after touching the animal.
It depends on the way of using the vaccine.
If such vaccine is to be taken orally, it will Haraam to consume it.
But if it is not taken orally but injected in the body or dropped in the nose with out letting it go inside the stomach, then it will be allowed.
Anything taken from a pig is Najis and absolutely forbidden to consume.