There's no reason for the FED to stop, or even slow printing of new dollars, "fiscal cliff" or no, though, so I think we can count on it's price in fiat continuing rise, too.
Actually it could. Suppose the theoretical case where the fiscal cliff result in a balanced, if not positive, budget. Then the FED could stop printing.
But yeah, not going to happen. There might be some tax increase/cut spending, but not as much as people think.
Well not exactly. The FED expansion or contraction of the money supply can be completely independent of the deficit of the govt.
When the Federal govt has an imbalance (receipts < projected outlays) it issues debt just like any company or person does. People, companies and countries purchase the bonds and pay for them in Federal Reserve Notes such that Federal Receipts + Funds from Debt Sales = Current Outlays.
The US govt can issue all the debt it wants (or issue no debt at all) and it doesn't require changing the money supply a single penny. The Federal reserve is a private cartel of banks who changes the money supply for the stated goal of "controlling inflation" and "maximizing economic output". Those goal (IMHO) are dubious but they exist and the FED will expand or tighten as needed regardless of what the federal govt does.
TL/DR:
The federal reserve is a private cartel of corporations and just about as federal as federal express. The only difference between federal reserve notes and disney dollars if the former is issued by corporations who has the forsight to get the Congress to bestow it wil rights as legal tender.