Once the MOSFET is turned on (Vgs > Vth) will Rds(on) vary with the increase of Vgs? Or it is not affected by the gate voltage once the MOSFET is on.
There's not much Rdson variation. You can get a sense for how much this changes by looking at the VI curves on your particular MOSFET's datasheet. For example, take a look at the IRFP260N:
These are the current vs. voltage characteristics for various gate voltages. The IRFP260N has guaranteed Rdson specs for 10Vgs; it's not a logic-level FET and expects 10V to fully turn on.
MOSFETs have two basic operational modes. If you want to operate them as a switch, then you want the current to be low enough so you're operating on the Rdson side of the curve: Vds = Rdson * Id. For a given gate-to-source voltage, there is a current limit above which Vds just shoots upwards because the MOSFET acts like a current sink. This is great for linear amplifiers but bad in power circuits and you usually won't want to operate here.
If you look at the datasheet curves, you'll note that the current limit changes quite a bit with Vgs. You'll also note that for the most part, the Rdson part of the curve does not change much with Vgs. At 25 C, Vgs above 5.5V has basically the same Rdson behavior, and at 175 C, Vgs of 4.5V or more has basically the same Rdson behavior.
Will it be affected only by the junction temperature?
Variation vs. junction temperature is fairly predictable and will also be in the datasheet. You typically see a factor of 1.5 - 2.5 increase from 25 C to the maximum operating temperature (150-175 C) and need to plan accordingly.