A backpack solves all of those issues. None of that is necessarily very large.
As for the shower, some work places have showers. Depending on how much you work out and what the climate is like you may or may not be a heavy sweater after a ride to work. People who work out often tend to sweat more because their bodies adapt to their routines and start to sweat at the slightest hint of physical exercise. People who don't work out extensively often tend to be more reserved when it comes to sweating. You are going to work, not a date. So it may not be an issue at all.
I can't wait to get a bike next season and start biking to work.
yes! i just have a milk-crate, but it makes all the difference.
yes also to OP on the "bicycling to work wakes me up super well" - i did in a northern US city for 6 years, and the icy/cold months when i had to rely on transit instead were remarkably more challenging, especially the mornings, even if i kept up other indoor exercise.
All bodies are different; I read the GP and thought, "yeah, no, I'm in fairly decent shape, and I have to work really hard for a good long while to break a sweat". I never have to use chalk while rock climbing, for instance. I think my body's adaptation at this point is that light physical exertion isn't even a warmup for me any more, and I've burned quite a bit of "insulation" off (biking in to work does make the fridge cold labs feel warmer). That being said, having a shower is another good way to wake myself up, sweaty or not.
WRT Chalk & climbing, propensity to sweaty hands differs from person to person, but for both myself and my climbing partner, need for chalk is almost all mental. When I've got a cool head and smooth moves, my hands are dry as a bone. When I'm freaking out on lead, my hands get progressively more slippery...
FWIW chalk improves the friction coefficient of your hands, and seems to reduce friction damage to the skin, so I try to use it anyway.
As for the shower, some work places have showers. Depending on how much you work out and what the climate is like you may or may not be a heavy sweater after a ride to work. People who work out often tend to sweat more because their bodies adapt to their routines and start to sweat at the slightest hint of physical exercise. People who don't work out extensively often tend to be more reserved when it comes to sweating. You are going to work, not a date. So it may not be an issue at all.
I can't wait to get a bike next season and start biking to work.