Must a coordinating clause always have a subject?

All English clauses have subjects. However, the subjects of clauses are often deleted, by various rules, if they are predictable from context or from higher or parallel clauses.

In example (1), there are actually two sentences; semicolons are essentially periods. So the clauses are not really conjoined. The second sentence is an imperative, as noted, and Imperative Formation deletes second person subjects. Note that if you did connect them with and, the sentence would be ungrammatical

• *You are getting yourselves into a very dangerous situation, and get out of there at once.

With true conjoined clauses, any material that's repeated in all clauses may be deleted from all but the first clause. Presuming a four-clause conjunction originally:

• You should wear a suit for the interview and
  you should wear a clean shirt for the interview and
  you should wear a tie for the interview and
  you should be punctual for the interview.

There are several rules involved. One removes all but the last and; another (Conjunction Reduction) is what's at work in (2), where it applies three times -- once each to delete you should wear from the second and third clauses, leaving only the objects behind, and once to delete you should from the last clause, leaving only the verb phrase, which is different from the VP in the first 3 clauses.

(3) is another example of Conjunction Reduction, removing the reference to Worcester (probably it) in the second clause.

But it's not only subjects that can be deleted; the following are Conjunction-Reduced sentences:

• Bill, Mike, and Harry worked on it all day.
• Bill rinsed, Mike washed, and Harry dried all the dishes.

A related rule, called Gapping, removes verbs alone:

• Bill ordered steak, Mike chicken, and Harry fish.