Do I have to get a Olympus version of the Godox TT600 which has HSS or are they all compatible across cameras?
The TT600 is a single-pin manual-only flash, so it's system-agnostic for the most part. There's only a Sony variant because the damn Sony multi-interface hotshoe isn't really ISO compatible and is a little too deep for an actual ISO flash foot to fit perfectly (sigh).
So, no, you don't have to get an Olympus version to get HSS with it; but weirdly, no, it's not compatible across all cameras; just with ISO-compliant hotshoes that aren't the Sony MI hotshoe.
Also, be aware, that HSS can
only be done through the radio transceiver that's built into the TT600; you can't use it in HSS on a camera hotshoe or through add-on radio triggers. If you need on-camera HSS/TTL, then consider getting a US$110 TT685-O instead.
Another "drawback" of the TT600 is that it cannot be firmware upgraded, like the TT685 can. I'm a Canon shooter, and Godox just posted a firmware upgrade for the TT685 models so that I can turn off the blinking LEDs that flash on the front of the speedlight to indicate when it's in radio slave mode. And to add the ID code feature when used with an XProC trigger (creates virtual channels so you can avoid interference with someone else using Godox X gear nearby). Both features work great. And I can't use either of them with my TT600 speedlights.
OTOH. It's hard to beat the pricetag. Particularly if you need multiples.
But I do tend to say that for a first/only flash, a TTL-capable one is probably a better bet. You can always turn TTL off on a TTL flash, you can't add it to a manual-only one.
Is this the same for the Godox XT1 triggers?
No. If you want HSS, the trigger has to understand the HSS your camera hotshoe is speaking, and that is proprietary and different for each camera system. So you definitely need an X1T-O (or, if they ever release them, an XPro-O or XT32-O) transmitter to get HSS with the TT600. Only the X1T and XPro can do TTL as well (say, if you decide to get a TT685-O instead of the TT600).