1. "Nehmen Sie von den zwei Tabletten hier dreimal täglich!" is not wrong grammaticaly, but implies another meaning.
For example, "von den zwei Tabletten hier" means that there are actually two types of pills. "Dreimal täglich" only means 3 times a day and does not give an amount (of how many pills have to be taken) and is therefore confusing. The information of how often is given, but not the information "how many" at once.
In your example, the "zwei" is put at the end because there is only one type of pill, and three times a day two of those should be taken at the same time. That way, the information "how often" and "how many" are better to understand.
2. Yes, the sentence can be rewritten like that. You have a very good understanding of German!
Please contact me anytime if this explanation does not help you.