[Christ]

Here's the problem with double clutching -

If you're doing it, chances are, you're doing it wrong. The point of double clutching is to allow the gears to match the engine speed that will account for the next gear. Most people just clutch, remove from gear, let off, clutch, put in gear, then hit the gas and let the clutch out like normal. It's a pointless 'third step' in that case, because the synchros still have to work, albeit slightly less.

When double clutching is done properly, you don't need to hit the accelerator until after you've left off the clutch pedal entirely.

It will not, however, damage your transmission. That dealer is throwing darts at an invisible dart board, hoping to sound correct. Ask them to show you proof that double clutching will damage a synchronized transmission, and if they can't put up the evidence, to stop spouting non-sense.

Other than wearing parts unnecessarily, such as clutch master cylinder, slave cylinder, and release bearing - again, negligible - TBH, it's just not necessary, and serves no benefit other than peace of mind in some people's cases.

Synthetic fluid isn't necessarily "better" for your transmission, but it is usually less viscous, so less engine power is necessary to defeat the impedance to movement that a thicker oil will cause.

Yes, you should ask the dealer to install it anyway, even if they make you pay for it (They most certainly will) unless they tell you that it will void any further warranty.

Synthetic oils have a longer life, better lubrication over the long haul, are more heat-resistant, and aren't made from something we could be using for fuel (not necessarily true, actually).