It depends on the signal generator. Some have a setting for what the load is - high impedance or 50 ohms. Others are only meant for 50 ohm loads so will give double the voltage when unloaded. Unless you can find an output impedance setting on yours it sounds like it is meant for 50 ohms. It will usually say next to the output socket. If it is an RF signal generator it is almost certainly 50 ohms.
If your sig gen is 50 ohm, and your load is high impedance that is to be expected. If your load is an oscope, you may be able to change the coupling from high impedance to 50 ohm.
Yes. As an example if the sig gen produces 10 volt:
for the case where the load is 50 ohm, 5 V is across the sig gen internal 50 ohm, and 5 V is across the 50 ohm load,
for the case when load is 10 kohm, then only 0.05 V is across the internal 50 ohm, and 9.95 V is across the load.