I'm quite surprised that lot isn't leaking or reeking- the long drops from the sinks mean there's a lot of water trapped in the pipework.
I'm also surprised the thing hasn't been back-syphoning for ever- if you have a read of the dishwasher installation manual it'll almost certainly advise a 600mm or so rise for the waste pipe.
The drain pipe was 600mm high - it was above the bottom of the sink originally, although I have it a bit higher now.
The main reason for trying to do something about all this was smells yes, but from the dishwasher not the sink. It has not always been smelly. Perhaps as the pipework has gunged up in the traps below the sink it has got worse. If the dishwasher has been syphoning mucky water back out of the sink trap into the dishwasher to sit in the bottom then a mucky sink trap would cause smells in the dishwasher.
I usually disassemble the whole lot and scrub it clean and reassemble it every year or two, this time however it was pretty bad so as I was buying new pipes to reconfigure the piping I've ended up just replacing 90% of the pipework with new stuff.
Your tee into the horizontal idea should work (in terms of breaking the syphon) and hopefully the water held in the dishwasher waste pipe will seal any drain whiff coming back from an untrapped attachment.
Yeah, that's what I've done:
The dishwasher is now connected on the final slightly downhill run and this is working perfectly, despite the excessively deep trap. There is a good air break at the point of connection - so good in fact that if I leave the dishwasher disconnected I can drain the sink without water even coming out the open hole - which I've deliberately placed facing upwards. Even while draining the sink there is still some air in the top of this pipe run.
The dishwasher doesn't need to be connected before the sink trap - when it stops pumping out an air break immediately forms at the point of attachment to the horizontal run and the contents of just the dishwasher drain hose syphons back into the sump of the dishwasher. It's below the level of the filter but if you remove the filter there is a good 2-3 inches of water depth after this syphon back - the dishwasher is providing its own trap so there shouldn't be any drain smells. (None yet anyway)
Putting it after the sink trap means no more gurgling or noise through the sink when it pumps out (no noise at all, it was quite noisy before) and no dirty water from the dishwasher sitting in the sink trap. There is basically no inter-mixing of sink water and dishwasher water at all now.
Ideally you'd be shortening the whole sink trap arrangements so the only permanently wet bit is the trap but that could be a right faff.
Yeah - if I could find a U pipe instead of the J pipe depicted in the photo above with the right fittings I could shorten the runs from the sink bowls a few inches and still line up with the level of the exit pipe making the trap less deep - but I've been unable to find a suitable fitting so far as all the kits I've looked at come with unequal length J sections. That is certainly a further improvement that could be made.
It's not until now that I've realised just how many different incompatible variations of pipe fittings there are for drain pipes - it was quite a challenge trying to find a way to connect the T that I ended up with as none of the readily available fittings are designed to do this. That T is actually one I removed from the original sink that was ripped out when we moved into the house years ago.