It seems to me that they launched that with a much higher TWR than normal...the rocket seems like it accelerated much faster than I'm used to seeing. Is that just the Russian way?
The Proton-M's initial thrust-to-weight ratio is about 1.5, which is somewhat high. Falcon 9 is about 1.2, and Delta IV Heavy is about 1.25, for comparison. And of course since the Earth subtracts a constant 1, that means the Proton-M's initial vertical acceleration is 2-2.5 times higher than those examples.
I'm not sure why it's different. Higher TWR is more efficient (to a point), but it also means there's potentially additional capacity to be had by carrying more fuel. Maybe it has something to do with Proton's origins as a gigantic ICBM, although that was a long time ago.
On a tangent, reading the Wikipedia entry for Proton-M, there have been 117 total launches and 11 failures. That's a mildly terrifying failure rate!
It occurred to me that with the Russians launching out of central Asia, there must be some chunk of land to the east of there with a ton of rockets falling on it, since even with a successful launch the first stages are just dropped. I couldn't find much info on that, but there are some really cool pictures here:
I don't think the payload matters much. Certainly its own weight is insignificant compared to the rocket as a whole. I'm not sure if they might load less fuel if the payload doesn't need full performance, but if so I think that would be a relatively small effect as well.
I didn't find actual TWRs listed even just for the launchers, but you can find the liftoff mass and thrust pretty easily, and just divide.
