Can a Yongnuo 565EX (no HSS) be used as a second flash (slave) with another flash that has HSS?

You'll have sync issues with any non-HSS flash when your shutter time is shorter than your camera's flash sync speed. The flash will fire when part, but not all, of the frame is uncovered by the shutter curtains. Only the part of the sensor uncovered by the shutter curtains when the non-HSS flash fires will receive the benefit of the non-HSS flash.

The answer to your question is no, if you choose a shutter speed that requires HHS, a flash that does not support it will either not fire, or it will light up only part of the photo.

This does not mean that your old flash is not useful anymore though, it is very rare that someone needs HHS or TTL on multiple flashes. Most people save a few bucks by buying second and third flashes without it. Large sites for strobists, like strobist.blogspot even recommend simple manual flases to start with:

http://strobist.blogspot.dk/2006/03/lighting-101-traveling-light.html https://mpex.com/compact-off-camera-flash-jump-starter-kit-with-flash.html

... can the YN-565EX (my old flash) be used as a second flash (for example, to light up the background for portrait photography),

Yes. The YN-565EX can be a "smart" optical slave to the YN-568EX as a "smart" optical master. Keep in mind, neither of these flashes have built-in radio triggering, and if you would prefer to use RF (radio frequency) triggering rather than optical triggering, which can be overpowered by bright sunlight, you'll have to add TTL/HSS radio triggers to the feet of both flashes.

... when I use HSS with the D7200 and YN-568EX combo?

No. You cannot use HSS with the YN-565EX as a slave flash, because it still cannot do HSS. All three links in the chain: the camera body, the triggering system, and the flash, have to be able to do HSS for you to have HSS. It will work as a TTL slave, and should fire in sync with remote power/zoom control, though, if you use CLS "smart" optical triggering or add-on Yongnuo YN-622 radio triggers.

Or you could go with Godox and get HSS.

In the Godox 2.4 GHz system, the TT600, the Godox analog to the Yongnuo YN-560 III/IV models differs in two big ways: it can do HSS as a radio slave to a TTL-capable Godox transmitter. And it can work in concert with Godox's TTL-capable lights. It costs roughly the same as a YN-560IV.

All of Godox's speedlights have built-in radio transceivers so can be either radio masters or slaves. But you'll only get HSS/TTL if the on-camera master unit is TTL-capable. So you could get a TTL-capable TT685N and a manual-only TT600, and have HSS with both lights, and still have TTL with the TT685N if you use both as slaves to a TTL-capable transmitter unit, or you slave the TT600 to the TT685N on-camera.