How could monetary policy be affected if cash "gifts" to consumers were more frequently used? A few years ago in the US, everyone who filed taxes received a check for $1000. 

If this method of inflation was logistically more feasible, how could this be used in monetary policy, and what existing monetary controls would it affect?
This is only inflationary if the monetary authority monetizes the borrowing. If the Treasury borrows $1000 on the open market by issuing a bond and sends the money to a consumer, no money has been created— it has simply been borrowed from one person and given to another. However, if the monetary authority purchases that bond (swapping base money for the bond), then it is indeed inflationary, as the amount of money now in circulation has increased. 

As far as its role in monetary policy— the decision of whether to monetize fiscal authority borrowing is part of standard monetary policy. The phrase you'll often hear is, "the monetary authority moves last," which reflects the fact that the monetary authority can either accommodate spending by monetizing it, increasing inflation, or do nothing, which has the effect of increasing interest rates on government borrowing.