Absolutely. This is a standard practice with pharmacies, and it is not at all unusual.
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Your new pharmacy will simply call your old pharmacy and transfer the prescription.
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The only thing that will be different is that you should give the new pharmacy a little extra time to fill the prescription. In other words, you might not want to wait for it to be filled the first time -- drop it off in the morning and pick it up in the afternoon or the next day.
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But with that exception, you shouldn't have any problems.

Of course but be aware that in some states you can only transfer it one time. This means if you take a medication with 4 refills and you transfer it you will have no refills left. So keep that in mind.

Of course. You have the information, go for it.