Definitely see what the transaxle fluid levels are. If it is leaking a lot of fluid, you should see a significant amount of fluid loss in the transaxle. Make sure the car is fully warmed up, running in park + e-brake up, level ground as possible. The level should be between the two sets of notches on the transmission dipstick, preferably closer to the sets of notches, as those are for check fluid levels on warm transaxle fluid.
See if you can clean that area up a bit - don't use really strong solvents, don't want to risk eating into that rubber boot close the transaxle - depending on how it looks, might be able to use a shop rag and a little solvent and wipe that area up. See how much leaking there is, lay down some cardboard - look for drips.
Assuming that the right CV joint is bad, if there is a clicking noise when you turn only left or accelerating while turning left - that might indicate a bad CV joint. Sometimes it can make a ticking, clacking, or knocking noise - depending on which way you turn, indicates which axle is under load. Ex. Turn right, makes noise - left CV are likely worn; turn left, makes noise - right CV worn. Sometimes it will make noises regardless of which direction - but should still make more noises one way vs the other. If they make the same noise regardless of which way you turn the steering wheel - could be both axles bad or you might have issues with the suspension / steering components.
Not out of the question for this sort of mileage. Usually, it is the rubber boots that go bad first, then dust and moisture eventually work their way into the joint, destroying it. If it is the axle, they can replace the seal at the same time and save you the additional labor. Usually cheaper to replace the whole axle - reman parts start around $40-$50 for an axle assembly, seals run under $10 (just from a quick spot check of the prices on Rockauto). Doesn't have to be done on both sides, so you can shave a little money there. Depends on the condition of left side.
Definitely get a second opinion from a qualified shop - if they all say the axle is bad, might want to get that replaced sooner than later. That sort of problem will not go away and only will get worse over time.