Your main panel has 2 separate bars that attach to separate 120V feeds. By attaching a single breaker to one of these bars, referred to as "buses", you get 120V. To get 240V, you need a double breaker that attaches to both bars at the same time. When you install this double breaker, it needs to fit so that the plastic separator tab along either of the hot buses slips down the middle of this double breaker. Is this device a welder or compressor? If so, you only need 2 conductor cable to connect to the double breakers. If this device is a stove, then it may need 120/240V, in which case you need 3 conductor cable. The black and red wires will connect to either half of the double breaker, and the white wire to the neutral bus of the panel. That's where all the white wires and some of the bare copper wires may connect to. If the device is 240V only, you won't have a neutral wire. The black and white will only go to the breaker. The bare ground wire goes to the ground lug, where all the other bare copper wires attach to. You will need 6/3 or 6/2 cable to handle 50A. Since 6/2 is rare, you may need to use 6/3 and ignore the white wire if the device is 240V only. Mull this over and then ask more questions.

"1) Is this the correct way to do this? " Maybe, maybe not. You say it is a 240 circuit, but the majority of ranges are 240/120. A 240 circuit has no neutral. Prior to 1996 240/120 circuits for ranges were OK to run with SEU cable originating at the main service panel. SEU has 3 conductors with the bare, concentric outer neutral bonded to the range cabinet for grounding purposes.. If this is the case, and your range is 240/120 than no, it is not the correct way to do this, code prohibits that circuit from being extended. The correct and legal way is to run a new, 4 wire circuit and install a 4 wire receptacle and a 4 wire cord on the range.

It is possible to make this change, but I don’t see the effectiveness in this. In your case, I advise you to reach out to a specialist. I know what I am talking about since I confronted a similar situation as you. Good thing my brother is a certified welder and was able to show and explain me the situation in my words, so I didn’t have to replace everything by myself and make things worse. He then fixed everything, and I have never had problems since then. That’s when I understood that https://weldingnearyou.com/school-cost/ was a good investment, and the money our family put in there shouldn’t be pitied.

I wonder is there is not some confusion among the comments here. Certainly there are appliances that use ONLY 240 Vac connections, without the neutral. Otherwise NEMA L6 and 6 plugs and receptacles would be in violation. Note that these circuits also need to be grounded. From the appliance point of view, the ground should be via a green insulated or bare copper conductor, but may be carried via armored conduit (raceway). Note that the other appliance conductors may require connection using devices temperature rated the same as the appliance conductors. This is a typical reason for hard wiring, because wire nuts and certain other devices are rated to 110C, and most plugs and receptacles only to 90C. The OP needs to determine the current requirement of the cooktop, needs to ensure that the conductors from box to cooktop are capable under the usual conditions of that current, and that there is a breaker protecting those conductors rated for that current. Then, to achieve GFCI, that breaker has to be replaced by one that is suitable for the box, or more precisely, acceptable to the local electrical code enforcement officer, who has final authority. How does the OP know that there are not modern breakers that fit his breaker box? This entire subject just begs for involvement of an electrician who will know all this stuff -- one hopes. To the OP, if something is shorted, an old non-GFCI breaker (if functional) will open the circuit. GFCI breakers are to protect humans from cases where there is leakage to some part one can come into contact with. Combo breakers (GFCI + AFCI) also detect arcing that might not be sufficient to otherwise cause a current overlord. These are mainly useful against house wiring subject to nail penetration in walls, and zip cords that are under rugs or otherwise subject to damage. If the feed to the box does not have a grounding conductor, then for use where the appliance does not need a neutral, the white neutral conductor can be re-purposed at the appliance box and at the breaker box by using it as a grounding conductor. Green tape will be needed to reveal that to future electricians. Last, let me observe for the OP that the drawing is defective and hence misleading. A cable is not a conduit and has to contain a grounding conductor that terminates in the box where the cooktop flex terminates. The flex, box, and grounding conductor have to be in good electrical contact.