Thru Axle and Quick Release Fork/Frame Compatability? Without using manufacturer specific parts:

Can I use a wheel with a thru-axle hub on a standard quick release fork or frame dropouts (with an adapter)?
Can I use a wheel with a quick release hub on a thru-axle fork or frame dropouts (with an adapter)?

We may also want to consider that there are also different thru axle sizes, commonly 12mm, 15mm and 20mm.  I think the most common size of thru-axle currently is 15mm.
Simply for your two points:


For a front wheel you can buy or machine a part that will allow you to run a smaller axle on the front than the hub is equipped with. This an adapter to run a 15mm TA hub in 9mm drop outs.




The rear is more difficult as hub sizes get wider with larger diameter thru axles. You couldn't make a 12x142 thru axle hub fit in a 10x135 QR dropout even if you shimmed the space at the end as the chain line would be off.


In terms of frame/ fork design, the design form for through axles offers no backwards compatibility with QR drop out design. I haven't seen and can find no evidence online of an adapter that allows you to run a 9mm hub in 15mm TA drop outs. That being said it could be possible to engineer something that created the required tension to hold the hub in place, but I wouldn't trust a home made solution.