Are there roadside bathroom facilities in Austria and Czech Republic?

If you mean something like this, no: https://previews.123rf.com/images/photoclicks/photoclicks0701/photoclicks070100025/731954-a-young-japanese-woman-in-an-open-air-hot-onsen-bath-in-the-snowfield.jpg

If you are talking about toilets, yes. All Motorway Service Areas (Autobahn Raststätte) have toilets.

Here is a brochure with a list of all Raststätte in Austria: http://www.arboe.at/fileadmin/uploads/ARBOE/kat/reise/tanken_rasten_2012.pdf
On page 10 there is a map showing them all. Other than on Autobahns, it is probably best to look for fuel stations, most will have toilets.

We had no trouble finding facilities in this region on a road trip last September; as others have said, gas stations and commercial rest stops. Many have a coin-operated turnstyle at the entrance, so have some small change in your pocket. We noticed that those with a modest "entrance fee" were most likely to be well-lit and clean.

What Chris was implying, but didn't say directly, is that if you need a toilet in Europe, you should ask for a toilet. I know it doesn't sound right to people in the US, but euphemisms like "restroom," "washroom," or "bathroom" may not be understood.

Yes...plenty! Europe is set up very well for road warriors. Most gas stations are part of a larger complexes with food, a gift shop, manned restrooms, and a playground for the kids. In the Czech Republic they are fewer and farther between but you'll find them. There are quite a few more in Austria. Once you get onto secondary roads it's a little more difficult to locate them. In those areas, just use the restroom whenever you get the opportunity, even if you don't "need" to.

An unfortunate and emerging trend is one of a coin-operated turnstile on the toilets, which gives you a voucher that can be redeemed against a purchase in the shop of the gas station. The days of carefree free peeing are behind us.