Can I write many verbs after "Do not". For example:
Do not eat, catch and buy blue crabs.
Is the above sentence correct?
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Yes, that's perfectly good, though it would be more natural with or rathr than and: your sentence suggests that it is telling you not to do all three things, rather than not to do any of the three things.
Almost all contexts that accept a verb will accept two or more coordinated verbs in the corresponding form:
I will wait and ask him.
He is eating and sleeping.
They have washed and put away the dishes.
Don't talk or move around.
Note that it depends on the particular verbs and the particular construction whether an object applies to one or all the verbs - sometimes it is actually ambiguous.
It is not well worded.
Take the sentence: "Do not catch and eat blue crabs." It is correct, but it might be taken to mean that only the combination is prohibited - and so you are permitted to catch them without eating them or to eat them without catching them.
Similarly, "Do not eat, catch and buy blue crabs" is badly worded. Your intended meaning is better expressed with "or" rather than "and":
Do not eat, catch or buy blue crabs.
But the answer to your main question is "yes". "Do not" can be followed by multiple verbs.