TL;DR: Yeah, you guessed right. You can't get HSS or TTL cross-brand.

Flash hotshoe protocols are mostly brand-specific

TTL and HSS are hotshoe protocol-specific. That is, you need a Canon camera to do HSS with Canon HSS-compatible flashes; and with a Sony A-7, you'd need a Sony HSS-compatible flash (i.e., probably a Sony HLV speedlight, or 3rd-party flash that says it does HSS on the Sony hotshoe). And that's just for HSS with an on-camera flash. Doing it off-camera is even tougher, because that communication protocol has to be duplicated by the triggers.

Sony's Multi-Interface Shoe is Weird

If you look at the foot on a flash and the hotshoe on the camera, the first thing to check is that the physical connectors match. The A7 is using Sony's new ISO-compatible hotshoe, which is unlike every other hotshoe out there. It does follow the ISO standard for hotshoes: the physical dimensions are correct, it uses the rails as ground, and the sync contact is in the center of the "square" of the hotshoe, so any flash/trigger that's ISO-compatible will work to fire the flash correctly in time with the shutter for exposure. But Sony is using a small series of contacts on the edge to do all the TTL/HSS/camera communication other than the sync signal.

This is unusual, but they are using this interface not only for flash communication, but also as an interface for other add-ons like microphones, GPS, Wi-fi, electronic viewfinders, etc., which is why there are so many different connectors (see the Wikipedia article on the multi-interface hotshoe for more details).

Nobody's Hotshoe is Just Like the Others

But they're more similar to each other than Sony's.

On all the other camera brands, the proprietary communication is done with additional pins/contacts on the bottom of the foot, next to the sync signal. Pentax and Nikon's pin pattern is one above it, and two below. Canon uses four below in a small square pattern. Fuji, and Olympus/Panasonic use the same pin placement as Canon, but of the four Canon signals, Fuji X doesn't have the lower left contact/pin, four-thirds doesn't have the upper left contact/pin. Everybody, however, puts out different signals across those pins/contacts.

For example, Canon, Pentax, and Nikon still do analog signaling (backwards compatibility with film-era gear), while Four-thirds signals, however, are digital. And even the film-era signals are quite different:

Pentax:

Nikon:

Canon:

And Nikon's quench signal goes from high to low voltage, and Canon's quench goes from low to high.

So, the only thing that can be communicated across all ISO hotshoes correctly is the sync signal. Everything else is black box potluck and good luck to the reverse engineers.

Radio Triggers Gotta Match the Hotshoes to do TTL/HSS

The triggers you mention are most likely to have the Canon or Nikon pin/contact layout, which means none of those additional signals come anywhere near Sony's contacts. And given how many contacts Sony has and the chances that digital signaling rather than analog are being used, reverse-engineering the protocol is probably a lot more difficult. The physical placement of Sony's contacts also probably means retooling or more expensive parts, so added manufacturing cost, which may put it outside the range of usual production costs for 3rd parties that get by on having low-cost products. As far as I know, at this time, there are no TTL/HSS-capable radio triggers for the Sony multi-interface shoe, and no 3rd-party TTL cables.

If you want HSS off-camera with an A7, likely you can only do this with a single flash on a Sony TTL cable. And for sports/action, you probably won't be able to find a cable long enough. The only radio trigger that may work are the Aokatec-TTL triggers, which are kind of like bargain RadioPoppers. So, you need a master unit, and since the A7 doesn't have a built-in flash let alone one that's a master, you'd still have to get a Sony HLV speedlight. It's not like Yongnuo makes a YN-622S trigger.

Also, given that you are action/sports shooting, considering the A7's (well, mostly mirrorless overall) slower AF, 5 fps burst speed, how Sony's mostly delivering slower lenses for E-mount full frame to reduce lens size (i.e., there's only a 70-200/4, and probably will never be a 70-200/2.8), and the lack of TTL/HSS 3rd party flash triggering, you may want to consider whether going Canikon for this type of shooting might not make more sense.

Fast action with longer lenses is the one big hole in mirrorless capabilities at the moment, and only the Olympus EM-1 really seems to have tackled it. And 3rd party support of TTL/HSS for mirrorless is another hole. For what you want to do, nearly any mirrorless camera would be the wrong tool.