This sounds a lot like a faulty ignition barrel or an electronic key that has a semi-flat battery. A worn barrel can often prevent the necessary contacts meeting to complete the ignition circuit and, likewise, an electronic key without sufficient power can also give these symptoms as it struggles to enable the ignition. This can be a very hit and miss thing, which is just how you've described it.

So check the condition of the battery inside the key and perhaps the condition of the car's battery as well. If the situation gets worse, a trip to the auto electrician is probably in order. At least your STS model has a relatively conventional ignition key set-up, rather than the push-button, proximity key of the SSS model which brings another layer of complexity to this function.

Changing a clutch in any car is a big job and can easily cost the sort of money you’ve been quoted. And when that car is a front-wheel-drive vehicle, there are a lot of things to remove (like the driveshafts) before the gearbox can be removed and the new clutch fitted.

While I agree that the symptoms you’re reporting do sound like a worn out clutch, I’d like to know what else the mechanic thinks will be wrong. He or she may, for example, be budgeting for the removal and machining of the flywheel as part of the clutch replacement, That can easily add a couple of hundred to the bill. Also, where is the mechanic sourcing the new clutch? You may have found a replacement kit online for the $500 you’re quoting, but is it a quality part from a reputable brand or a no-name piece of rubbish from an internet clearing house?

Try gently releasing the clutch while you’re holding the lever in the reverse position. It’s probably an adjustment issue, it could be a clip that’s broken, and nothing serious. I would take it to a mechanic who should be able to tell you exactly what is wrong with it.