"From that entrance, please."
"Douzo" can be applied to many situations, so when giving someone something douzo is used sort of as a "here you go, please take this", but in this case it's being used to indicate to "please use that entrance". What it means has to be understood by the context. The literal translation of this sentence would probably be closer to "that entrance, please". It's sort of a request.
Douzo i always imagine like someone pointing their hand towards something for "please"
So "sochira no deguchi douzo" is someone gesturing an open hand, guiding to the exit, go there please
Whereas "sochira no deguchi kudasai" is someone pointing at that exit saying give me that please
I don't know why you were downvoted, but your first assumption is correct.
そちら is the polite version of そっち, but also of それ.
それ+の = その
polite version: そちらの
And given the phrasing of the sentence, we can assume it's for example a staff member guiding a costumer, and therefore polite speech would be used.
"From that entrance, please" is a little vague, but it essentially means "Don't enter here, use that entrance over there instead." It could also be used to answer the question "How do I get into this building?" where the most obvious entrance may be blocked or otherwise obstructed.
From my experience, usually ください is attached to requests for the listener to do something. It seems like the speaker is directing the listener to do something here, and thus "please" would be ください. Its mentioned below, but どうぞ also has more of a "have this" feel and ください also has a "give me" feel, but I've seen ください most often with "commands". (行ってください - please go)
ください is used when you are asking for something and both you and the listener are roughly on the same social level.
You should use ～ていただけませんか and similar constructions when speaking to someone who is "higher" than you (i.e. your boss).
Second usage of ください is when you want someone to do something that is beneficial to him - for example この薬を一日二回飲んでください, regardless of your relative social status.