"In accordance with the requirements of Russian legislation, the Rules of Internal Labor Order have been amended."
I agree the first "the" needed removal - we don't know which specific rule (which "the" implies). However, "the Rules..." are known: the rules of the company.
"Starting from March 1st 2015, advance payments will take place on the 21st of each month (instead of the 20th)."
Since it's on a regular basis, use "advance payments" in general and just simple future form. "Will be taking place" sounds like one specific date, not a change to the schedule. I don't agree with the change to "monthly" - it sounds confusing ("monthly what?").
"Employees making recurring payments and transactions are recommended to take this information into consideration."
Just the order of phrases needed fixing.
Apart from those quibbles, your translation still makes sense - it's actually quite good.