I commute on a bike that's several levels below a cheap $300 Walmart one -- and it's glorious. Every day, I get to work in a better mood than when I left home. I defy anyone who drives to work to make the same claim honestly.
I agree completely -- 5 days ago my bike commute took me through 6 inches of snow. The day after, -25°F (-31.6°C) windchills. Your body gets used to the cold if you bike every day as winter sets in. There really is nothing like putting on the thermal underwear + face mask and pedaling home. Keeps one's spirits up during these short winter days.
If anything, hot summer days are harder. After all, you can always add layers if it gets colder, but public decency suggests that removing layers caps out at shorts and t-shirt.
I build myself a fixie (http://en.wikipedia.org/wiki/Fixed-gear_bicycle) from an old bike last summer and I now always take a longer way home because it's really fun to ride. The bike did cost me under 100 euros :)
Northeastern rustbelt city. Hot and muggy in the summer, freezing cold in the winter (it's raining today but last Thursday and Friday were baltic), soggy the rest of the year.
Oh, yeah: and hilly. And there are almost no bike lanes.
So shower when you get to work instead of before leaving for work. Or, don't bike. There is no magical thing that is going to prevent you from sweating.
Bike or car, I can only conclude that you enjoy being at work more than being at home. I can't think of a single incentive that would cause me to arrive at work in a better mood than when I left home.
Home is where my "life" is. Work is where... well where I work