Is It Possible for a Country to have a Gold Standard without a Fixed Price of Gold?

(1) Is it possible to have a gold standard without a fixed price of gold?

No, with some provisions. First, I presume here we are talking about the price of gold in terms of currency not its relative price (relative to other goods and services) which was always allowed to fluctuate. Gold standard by its definition requires that money is fixed to gold at some fixed rate. If that does not hold it would not be a gold standard. However, the fixed rate can be changed overtime. For example, government could decide that one day the fixed rate will be 10 USD for 1 ounce of gold another day it will be 15 USD for 1 ounce. But this is not the same as having floating price for gold.

(2) Are there any countries left today which tie their currency directly to gold or some other precious metal?

No. Depending on what you count as country, the late Islamic State in Iraq and al-Sham (ISIS) had rolled out a trimetallic currency (see Oxnevad 2016). Their currency was to be made directly from gold, silver and copper hence would be directly tied to those commodities. Although, by international community it was never recognized by anything more than a terrorist group, it would probably satisfy the Max Weber's definition of state during mid 2010s. However, that likely would no longer be applicable presently after they lost all their territory to Iraq and Syria.