I have on old MTB I ride to work on days where there is no safe parking. Some days I'm lucky and the Kindergarten is fenced or I can park inside. Then I take the nice bike. Its really worth having a bike that rides nicely, but is inexpensive and really boring to look at. If it does get stolen you aren't out of pocket more than $150 or so.
Also in summer sweat is unavoidable. I wear cycling clothes and just take along my work clothes in a bag along with a towel and baby wipes. The wipes take care of the sweat and you towel down for the rest. Only takes 5 minutes or so and all you need is a bathroom where you can get changed into clothes you can wear for the rest of the day.
On road bikes... they pass me on my mtbs all the time. I don't worry. I'm riding to my best ability and if they are faster, then I need to train more.