… can you use the YN-565EX (my old flash) as a second flash (for example, to illuminate the background of portrait photography),
Yes. The YN-565EX can be an "intelligent" optical slave for the YN-568EX as an "intelligent" optical master. Note that none of these flashes has built-in radio activation, and if you prefer to use RF (radio frequency) activation instead of optical activation, which can be dominated by intense sunlight, you should add TTL / HSS. Shoots at the feet of both flashes.
… when I use HSS with the combo D7200 and YN-568EX?
No. You can not use HSS with the YN-565EX as a slave flash, because you still can not do HSS. The three links in the chain: the body of the camera, the trigger system and the flash have to be able to do HSS for you to have HSS. It will function as a TTL slave, and should trigger in synchronization with the power / zoom remote control, however, if you use the CLS "smart" optical trigger or the additional Yongnuo YN-622 radio trigger.
Or you could go with Godox and get HSS.
In the Godox 2.4 GHz system, the TT600, the Godox analog for the Yongnuo Yong-YN-560 III / IV models, differs in two ways: it can make HSS as a radio slave for a Godox transmitter with TTL capability. And it can work in concert with Godox's TTL-capable lights. It costs approximately the same as a YN-560IV.
All Godox flashes have built-in radio transceivers, so they can be radio masters or slaves. But you will only get HSS / TTL if the main unit of the camera is compatible with TTL. Therefore, you can obtain a TT685N with TTL capability and a manual TT600, and have HSS with both lights, and still have TTL with the TT685N if you use both as slaves for a TTL capable transmitter unit, or slave of the TT600 for the TT685N in the camera.