I think FSR will get rid of lime build up, as it is a mild acid...but I believe it isn't strong enough to damage the gelcoat... Vinegar should also work fairly well..

since it will attack the lime fairly aggressively, and is also safe, if a bit smelly, for the gelcoat. BTW, I'd go with the vinegar, as it is dirt cheap. Just be sure to rinse the area thouroughly regardless of which you use.
As for the Lanocote. It is much easier to apply from the exterior of the boat, if it is on the hard... a bit more difficult to apply from the interior. What you do is take a blob of it on the end of a wooden applicator and apply it to the ball. Then work the valve, to spread it out. If you're doing this from the outside, it helps to have a drudge/partner/crewmember on the inside of the boat working the thru-hull handle.

If you're doing this with the boat in the water, you have to remove the hose from the inside of the valve...and then apply it...
As Patrick has pointed out... in most cases the amount of force behind the water is not all that great, and easily stopped....
What one friend of mine has done, but I have not tried, is to use a small sponge grabbed with a pair of hemostats to apply the lanocote. It is probably more effective than the stick in applying an even coating of the lubricant. And, if you size the sponge properly you can use it to stop most of the water flow from when you operate the valve handle.