A few things on the 600:They have a Nikasil bore which is very long lasting and rarely has problems at the mileage on yours unless it has been very poorly maintained. The Cam chain tensioner is also a rare failure because of its simple but very effective design
If you are removing the carbs remove the Rocker cover and check the chain for wear by the cam timing marks and you need to remover the alternator cover to line up the marks. There no gasket on the cover as it run dry. Tie it back while it is off. If the marks are off then it is the chain that's wearing.
Chains wear unevenly and the tensioner cannot take up the slack fully because of this. A bit of rattle from cold until warm is usually cam chain wear. As the cover is off check the valve clearances. They can be out in as little as 20000 miles or 75000 miles checking them rules them out or in as part of your issue. These bikes will still run fine with the timing out one tooth due to chain wear.
You can remove and dismantle the plug caps, there are notorious for corrosion within and many replace them with NGK SD05F. I cleaned out my own and replaced the internal resistors with ones for an old NGK set as they are wire wound compared to the Yamaha Carbon ones and less prone to failure, and the top and bottom boots of the caps with NGK one also. The internal spring needs to be cleaned is reusing the old caps and the tiny internal disc spacer cleaned of dumped as it corrodes badly and break easily. They spring seat needs to be cleaned with a sharpened screwdriver or something like it. Before refitting the caps cut 10mm of the ends of the wires before screwing them back on to the caps.
Carbs: Carbs can be a pain to remove or refit but loosen everything fully, as a rule of thumb loosen the front and rear bands until the bolt is flush with the end of the nut on the band. Removing the bolt completely in not necessary and if you do the spacer will drop into the most inaccesible place, this needs to be used as it prevent ove-rtightening the band and damaging the intake rubbers. The rear band has no spacers, but are seated in one position. It is worth while replace the intake rubbers due to the age of the bike. (Wemoto do a good set at a reasonable price) Coils need to be removed and Fuel pump also needs to be removed. Remove the side cover and the battery. loosen the 2 side bolts remove the top bracket and pull back the air box as far as it will go and tighten the two side bolts to hold it there. Move the clutch cable out of the way to ease removal. Loosen the cable completely and remove from there holders. Pull back the carbs from the intake and this can be extremely difficult as they will be stuck in place however perseverance pays off here. Cables are removed when carbs are half way out usually the left side remove the cables from the carbs.
When dismantling the carbs do not mix up the bits from each carb as the will have different wear and match the ones they cam from best. Replace the float bowl seals and remove the pilot jets and check them thoroughly. They have a tiny star web internally and makes them difficult to clean. Soak in carb cleaner for a few hours or overnight usually helps. Remove the main jets and check the O ring worth replacing if flattend or damaged. Replacing the float valve O Ring is also worth while to save have to remove the carbs again as they can go hard/perish when idle for long periods. I don't usually replace the needle valves as they are not wear prone and are better quality than most aftermarket ones, plus it save resetting the float level. Use ordinary grease on the float bowl seal as it will aid fitting and the fuel will wash it away easily if it gets inot the system
The TPS wears and due its carbon track design and is a hit and miss causing uneven idle and surging on low throttle. Be aware when buying new ones as there are two types that look exactly the same design with the same markings, one is spring loaded clockwise and the other is spring loaded anticlockwise. If the add does not specifically mention the FZS it may not fit. There is a Suzuki one the fits and can be half the price of the Yamaha ones new. Usually if the fits the right side of the carb and its connector is facing up and back it is the correct one. An example of this is the XJR1200/1300, it looks exactly the same, but is the reverse of the 600 and will not work.
You should have enough to go on here to start.