The best sentence that combines the two ideas using a coordinating conjunction is:
"My dog loves running on the beach, but she doesn't like seeing other dogs on the beach."
However, this specific option wasn't provided in your list. The closest option from your provided sentences is:
"My dog was running on the beach and she doesn't like seeing other dogs on the beach."
While "was running" changes the tense slightly, it is the most suitable choice that maintains the coordinating conjunction "and." The other options either change the meaning or don't use a proper coordinating conjunction.