Congrats on the Nishiki, Custom Sport.
Yes, you should replace the rusted cables. Do not replace the derailers, clean and lube yes. I use spray chain lube with the small tube they provide to get inside the rear derailer.
Brakes. Check the toe in of the brake pads. If they are not toe'd inward, toward the front of the bike, they will squeal, period (both front and rear). Next, if they still squeal, clean the sides of the rims with rubbing alcohol. If they still squeal after that, change the pads. However, I've never had to change pads to get rid of a squeal. I've changed them to improve the braking power, but not to get rid of the squeal.
As for the turkey wings, I don't share the desire to remove them. They stay on all my bikes that have them. They do not have good mechanical advantage so you won't be able to improve the braking to the same level as the main brake levers, but for scrubbing off speed while riding on the top bars, they are just fine. The only danger they pose is someone insisting on using them for a panic stop, which they are not suited for. It's your preference.
Shifting a friction shift unit is not like an index shifter. I'm betting you have a stem mounted shifter (probably Suntour, maybe Shimano). You have to overshift many of these units to get the chain to shift, then bring it back. I think, just my opinion, stem shifters require this type shifting more than downtube, but most of my downtube shifters require some overshifting. Important, check your chain stretch. If that chain is old and stretched, it needs to be replaced. A new chain, or one that is in spec. will shift better with less overshift, but it will still require some.
With regard to the colored rims, your best bet is ebay. If you want them cause that's what you like, go for it. If you want them cause they are cool, I've not seen any high end rims come in different colors. I think the colored rim thing is retro. But then you're riding a retro bike so yeah, it might work.
As for making it look badass. IMHO, there is one change that really improves the look of an old, retro bike to make it look more up to date and cleaner. Replace the brake levers with aero levers. Not only does it clean up all that wire mess near the bars, they are safer because aero levers improve your mechanical advantage when braking. Additonally, I've had the wires interfere with my hands a couple of times at inopportune moments.
If you have a chance, please take pics and post them on the Nishiki Serial Number database here on C&V along with the serial number. T-Mar and the rest of us would like to see your bike.
I have an 87 Custom Sport and really like it. I think they are an under-rated bike.
Good luck bringing it back.