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1. Are all these phrases correct, and do they mean the same:

a) "Have in mind..."
b) "Keep in mind..."
c) "BEAR in mind..."

(And if they are all correct, do they all sound just as natural?)

2. Is this sentence correct:

"I'll be ATTENTIVE to my e-mail box, waiting for your message."

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1. Are all these phrases correct,Yes and do they mean the same: Broadly speaking, yes.
a) "Have in mind..."
b) "Keep in mind..."
c) "BEAR in mind..."

We more usually say Remember . . .

(And if they are all correct, do they all sound just as natural?) Yes. Perhaps B is a bit less common.



2. Is this sentence correct:

"I'll be ATTENTIVE to my e-mail box, waiting for your message."

We are usually attentive to a person, not to a thing.

More natural is eg I'll keep checking my in-box.