Do people make puns out of the Earthly Branches in Chinese culture? For example, the earthly branches represent divisions of time and also directions.

Are these multiple meanings used in puns?
The individual branches can also have separate meanings from the branch system itself. For example, 丑 can also mean clown.  

Do these second set of non-branch meanings get mixed or played off or punned with the branch meanings? (e.g. The year of the Ox being called the year of the clown, it being clown o'clock etc.)
丑 can also mean clown


No, 丑 usually means ugly in Simplified Chinese. 小丑 means clown. 

We usually don't say 丑年. We say 牛年. 

I guess you could play with puns if you try to, but in a normal conversation people won't mix the meanings.