If you shower before you ride then you have twelve hours or so head start on bacteria - it takes them that long to multiply, excrete and cause a stink. So even if you are glistening with sweat at work then you can change your clothes, wash your hands and face, sit at desk, work, no smell needed. If you do stink then maybe it is because you lack fitness and basic hygiene, which is nothing to do with there not being a shower at work.
When was the last time anyone washed a seat in a car? Never! A car seat is one of the most disgusting places to sit. Would you not change the sheets on your bed for years? No. Yet sit in the same seat in a car, year after year, with a greenhouse around it, sucking in carbon monoxide from the tailpipe in front - no idea why people think that is hygienic.
You need to get practical!
Biking 15 miles to work is not practical in 100+ degree weather with humidity averaging between 60 and 90 percent. Moving to be within biking distance of work is not practical when jobs change every few years and there are two of us living in the house which is fairly central to both workplaces.
And I wash the interior of my car yearly - your phobia about 'dirty car seats' is fairly silly.
I'd say trying to not die from heat exhaustion or stroke while riding uphill on a bike in 100+ degree weather is practical.
That's around 37 degrees celcius for us non-US people =).
I'm from Sydney, and we often have low 40's (Celcius) during summer. We currently have raging bushfires over here in some areas =(, and it's easily 40 degrees.
I bike to work (and also for fun on weekends) during that sort of temperatures, and I've never been worse for wear...
I also jog regularly during that weather (although I find that harder than biking - less breeze).
Mate - you're not going to get heat stroke from riding your bike a few Km at 37 degrees, seriously...
As your fitness improves, you'll probably find it easier.
Just stay hydrated (you do carry water, right?), apply sunscreen, and you'll be fine. Sure, it's harder work than driving a car, but exercise is hard work (despite what those late-night TV ads tell you).