With your derailleur hanger, you'd need the direct mount version of the derailleur, not the claw or reverse claw. As for SGS, Shimano has three ranges of rear derailleur cage - SS is short, GS is medium, and SGS is long. This matters for some derailleurs that are sold in multiple variants. It looks like the RD-TZ500 derailleur exists in GS and SGS versions, but the RD-TY500 appears to be SGS only (though I could be wrong).
##does direct mount means donut form and claw means c shape?
I wonder thought whether the specs I previously saw for the yorkville are what you actually have. Is this your actual bike?
https://www.norco.com/bikes/2021/cit...lle-step-thru/
### yes im pretty sure it is this on. it has probably more than 12 years .
I just noticed the specs seem mixed up. The page for the yorkville (non step-thru) list the rear shifter as RD-TY500, but this is a rear derailleur part number, not a shifter. Both pages list the rear derailleur as the RD-M360, which is a higher grade part than the RD-TY500 derailleur I previously thought you had. Either derailleur would work fine, but if you could find it, I would get the RD-M360, or even RD-M390 as I suggested before. I think these are available in both GS and SGS versions so be sure to get the long-cage SGS for your 34t freewheel. But if you feel safer with the RD-TY500 since it explicitly says its for 6/7 speed, that should work fine.
The shifters are a bit of an open question. The yorkville's page lists the Tourney EZ-Fire ST-EF500 as the front shifter, while the step-thru page lists the Altus Rapidfire ST-M310 for both shifters.
https://bike.shimano.com/en-EU/produ...F500-7R4A.html
Either way, these are combined brake lever/shifters, so if you were to go with any of the levers I previously listed, you'll likely have to deal with replacing the brake lever too.
Fortunately, there appears to be a nice answer here. There's a 6-speed version of the tourney EZ Fire Plus shifter, the ST-EF41-6. This would replace your existing brake/shift lever. I didn't know EZ-Fire Plus existed in 6-speed (it's like Rapidfire, I don't know what's different). It turns out too that the SL-TX shifters I cited before, are fancier than I'd realized, having a a push lever for downshifts as I'd presumed, but also a push button for upshifts.
## having a motor on i purchased a pair of electric brake lever that cut electricity when pushed so it cannot be combine.
https://bike.shimano.com/en-EU/produ...T-EF41-6R.html
https://si.shimano.com/pdfs/um/UM-00...0R-ENG_FRE.pdf
You might still consider separate brake levers though - they give you more flexibility in adjusting their positions independently, but if you're happy with your existing brake/shift levers, the EZ-Fire levers can't be too bad.
As for where to buy in Canada, searching from here in the US, all I could find seem to ship from overseas. You might have better luck searching for yourself.
https://www.amazon.ca/Shimano-EF41-3...Y3E/ref=sr_1_1
https://www.ebay.ca/itm/Shimano-Tour...S/313372564708
https://www.amazon.ca/RD-M390-Derail...DSD/ref=sr_1_3
https://www.ebay.ca/itm/RD-M390-Acer...m/133690057120