The retort I had is that this is not posted anywhere, and even if it was what legal enforcement does the campus have against me. They can't find out who I am, They can't detain me. They can't hold me until police arrive. So how would they enforce that policy. After it escalated to that level what penalty for trespassing would there be, and can I even trespass on public property.
You were trespassing
The community college is a public institution but they can decide what part of their land you can walk on and in what circumstances. Just like the military is a public institution but they don’t let you walk across their shooting ranges.
To be clear, in the absence of clear “no dogs allowed” signage, you were not trespassing until you were told about the policy. At that point, you were legally obliged to remove yourself (or more precisely, your dog) from the campus as soon as possible. When you refused to do so, you became a trespasser.
It’s trivially easy to find out who you are. One photograph, one reverse image search they’ll know everything about you right down to your shoe size. Even if you don’t use social media, I’m sure some of your family and friends do.
In most US states, trespass is a misdemeanour and also in most states members of the public can arrest someone who is committing a misdemeanour in their presence. They can use reasonable force to do so and can hold the arrestee until they can transfer them to the custody of a law enforcement officer. Admittedly, this seems unlikely but it is possible.
If you have caused damage, you can be sued. It seems that your discussion with the college staff was somewhat protected so the loss of productivity of those staff members is a loss that the college suffered and that they could sue you for. Again, not likely but possible.
Alternatively, they could just report you to the police who may or may not bring charges.
Note: this assumes the dog is a pet. If it’s a disability assistance animal, it can’t be excluded.
It depends on who they are, and where you are. A property owner or "person with authority" can order you to leave their property. If you don't leave, you are trespassing, which is at least a misdemeanor, and a person who witnesses you to committing a misdemeanor usually can make a citizen's arrest (which means detain and remand to the police). But it depends on what the law is in your location. And it depends on who "they" are. Campus police would have authority to order you to leave, an accounting TA would not. It doesn't matter if this restriction is posted or not. You say that this is a "community college" which suggests that it it an arm of the government to some extent, so the rules can't be completely capricious (when on private property, you can be told to leave because they don't like you). Government operations are expected to operate in a more rule-like fashion, but the lack of any posting does not mean there is no rule. For example, dogs are prohibited in university buildings in Washington state, but this isn't posted at the U. Washington -- because that law is published, you are expected to know and comply with the law.
You'd have to name the institution, if you want a pointer to the relevant laws.