The first two sentences you have posted are correctly connected:

I adopted two dogs, which weigh approximately 100 pounds each. - There is a comma and the pronoun "which." Thus the sentence is properly formed.

The second sentence "I adopted two dogs, each of which weighs approximately 100 pounds." is also correct for the same reason.

A run-on sentence is a sentence in which two or more independent clauses (i.e., complete sentences) are joined without appropriate punctuation or conjunction. This is the case in the next sentences:
I adopted two dogs, they weigh approximately 100 pounds each.
Only a comma separates these two independent clauses from each other and there is no logical connection between them. These two clauses could easily make two separate sentences.

The same is true about the other two sentences.