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

… the YN-565EX (my old flash) can be used as a second flash (eg to illuminate the background for portrait photography),

Yes. The YN-565EX, as a "smart" optical master, can be a "smart" optical slave of the YN-568EX. Remember that none of these flashes has built-in radio triggering. If you prefer radio frequency (RF) triggering rather than optical triggering, which can be overwhelmed by bright sunlight, you need to add TTL / HSS. The radio will fire at the feet of both flashes.

… when I combine HSS with the D7200 and YN-568EX?

No. You can not use HSS with the YN-565EX as a slave flash, as this is still not possible. All three links in the chain: the camera body, the trigger system, and the flash must be able to perform HSS for you to have HSS. It works as a TTL slave and should be triggered synchronously with the power supply / zoom remote control if you are using the optical "CLS" trigger or the additional Yongnuo YN-622 remote controls.

Or you could go with Godox and get HSS.

In the Godox 2.4 GHz system, the TT600, the Godox analogue differs from the Yongnuo YN-560 III / IV models in two ways: HSS can be used as a wireless slave of a TTL-enabled Godox transmitter. And it can work with Godox's TTL-enabled lights. It costs about the same as a YN-560IV.

All Speedlights from Godox have built-in radio transceivers, so they can either be funkmasters or slaves. However, you will only receive HSS / TTL if the camera's master device is TTL-enabled. So you could get a TTL-capable TT685N and a manual TT600, HSS with both lights and TTL with TT685N, if you use both as slaves to a TTL-enabled transmitter unit or use the TT600 as a slave the TT685N camera.