OK the installation is fine, which is great. But the cables don't look very new, so I would suggest replacing them. For example, with these: https://www.amazon.com/JAGWIRE-ROAD-SHOP-KIT-Complete/dp/B00L2BQJM4
The SRAM Supercork bar tape on Amazon is a good choice too.
This is a good tool to check if your chain needs replacing: https://www.amazon.com/Park-Tool-CC-3-2-Chain-Checker/dp/B000BR3LHQ/
>Would this be a suitable replacement?
No, absolutely not. The shape of the cable end is wrong. (you need C but linked cables with A)
This is the same kit in a (C) version that fits your bike. The difference may look minor but it has the cable ends that fit your brake levers.
If your area has a "bike co-op" you can purchase only the cable and housing you need to save some money.
>Here's a link to the condition of my bike
Yikes, somehow those pictures all ended up as thumbnails!
That's a 1992 Trek 750: white with red decals.
It has (what look like plastic) low-profile cantilever brakes.
It had/has grip shifters.
Your wheel has a 7 speed freehub. It uses a cassette with 126mm spacing. Folks here on reddit may suggest that you've got a 7 speed screw-on freewheel, but they're wrong.
You can easily replace that wheel with any modern 8-9-10 speed 130mm wheel: you need to put a 4.5mm spacer between the cassette and wheel if you stay 7-speed. It makes it a little more difficult to change a tire since you're jamming a 130mm axle into a place spaced for a 126mm axle.
Notched 4.5mm spacers are better than plain 4.5mm spacers for this application but your options will be limited based on what country you order your gear from.
You can also replace those (plastic) low-profile cantilevers with any number of better performing cantilever brakes or even mini-v brakes that clear your fenders. Those smooth stud cantilever post brake pads are some of the worst brakes I've ever worked on. Holy shit, those things are terrible! Even a modern "bottom of the line" brake set is preferable!
You can drop in the SL-M310 shifter I linked above but it's narrower than the grip-shift you had. You'll either end up with a gap between the rubber grip and the shifter, or you can squeeze them together and the left/right will be a bit asymetric. Or you can buy both the left and right shifters at the same time. The asymmetry didn't bother me at the time but I didn't live with it long: it may not bother you at all.
If you want to swap the brake lever and shifter to a combo unit you need the 4 finger EF51. You must make sure you're ordering the 4 finger version and not the 2 finger version. Amazon sellers don't always get it right and you may need to return it if they send the incompatible 2 finger version. There's a little red plastic bit on the underside of the 4 finger version you need to move to the C position to make it compatible with your brakes.
Side note: that's a very versatile frame. It's identical in geometry to Trek's 560 touring bikes of that era. It's comparable to a modern "light-touring"/gravel/adventure bike. You've got clearance for some wiiiiide-ass tires, under fenders, on that frame. I've run 40c and had clearance but my fenders were too narrow to support something wider.
(I don't know the maximum tire capacity... because... fenders).
You can convert it to drop bars without running into geometry or parts problems like modern hybrids. It started life as a drop bar frame and that makes it a whole lot simpler to convert back from a flat bar to a drop bar than any other "conversion" you'll see in this reddit sub.
Trek took a touring bike frame, threw a flat bar on it and called it a "hybrid" while swapping out a road triple for a mountain triple. The touring bike versions have better braze-ons and tube butting.
Here's a link to the condition of my bike
In addition to the shifter hardly working (I had to grip it a certain way, or else it would just continuously spin), the cable connecting the shifter the the derailleur snapped. Would this be a suitable replacement?