Which raises more questions. What forces cause the lava to rise to the extent that it raises the plates?
It doesn't. Plates are moved
sideways by convection of the mantel. They buckle upwards where they collide.
That's the short version, anyway.
What causes magma to rise, at volcanos for example, is heat and pressure.
How far above the planet's mantel does the magma rise before it becomes cooled enough to harden into intrusive igneous rock?
Well, if you think about it, if you have a bit of magma embedded anywhere in the crust, it's surrounded by rock which is below the temperature and pressure needed for it to be ductile; and eventually the magma will cool to that temperature.
(This is a very slow process, because of the low thermal conductivity of rock, and so incidentally constitutes another proof that the Earth is not young; it would take millions of years for a really big intrusion to cool down to the temperature of the surrounding crust.)
If it is the cooled magma rock that is tested by the radiometric dating, does it date differently than the rising magma and if so, why?
You can't date magma.
To see why, consider as an example uranium-lead dating. This works by looking at how much uranium has decayed into lead, as hopefully you know. Now when rock is in its liquid form (magma) there's uranium and lead mixed all through it. But when it cools into rock it crystallizes, and for chemical reasons some of the crystals formed can accept uranium into their crystal structure but not lead. This starts the clock --- we know that when those crystals were formed, they contained no lead. Obviously this doesn't apply to molten rock.