Can one lower and raise the roof on a 2013 SL63. remotely with the key fob?

Yes.

Three things: (1) you can’t stand very far away; (2) the battery in the key needs to have lots of life left; (3) the remote operation of the top significantly extracts life from the key battery.

Why (2) and (3)? Just curious.
I mean, once the fob sends the necessary signal to the car, the top will be operated by the car's own battery, needing nothing more from the fob.
What do I miss?

What you say makes sense, but I’m telling you the way it actually is after many years of ownership- on this matter I cannot speak on the technical side.

If you notice with cars in general, most key fobs with lower battery levels do not lock/unlock a vehicle unless very close to the vehicle as opposed to the same vehicle locking/unlocking from far away after new battery installation. That’s one of the ways of noticing it’s time to change the key fob battery. Therefore I imagine that the key fob broadcasts a certain strength of signal which is directly proportional to the life remaining of the battery inside. Like a halogen flashlight with batteries maybe. Just guessing here. Either way, you can trust what I said above. Have had two R231 SLs and I just bought my third yesterday.

Congrats on the purchase - is that the '17SL63 I see in your summary - ? color .... Steve

Thanks!

No- the ‘17 SL63 in my profile is the one I’ve had for 5 years! It’s brilliant blue. The inbound SL63 is iridium silver, also a 2017 but loaded with every option known to man and super low miles. I happened to stumble upon a gem and decided to have a spare. Lord knows a hardtop convertible Mercedes like R231 will never again be born on our planet.

! Well Done !
Agree that this package is hard to match, much less beat !
Enjoy,
Steve

Thank you!

I should have mentioned- my second R231 was a wonderful Designo Diamond White Metallic 2019 SL450 which I ordered from the factory with ABC suspension and various other goodies. Like a shnook I sold the car after 8 months (traded it in for another car) — lost about $20k.