Motors are special
A "naked" motor like yours (i.e. not part of a listed appliance) falls under a special set of Code provisions known as Article 430 due to how common "naked" motors are in industry and the peculiar nature of motor loads. Many provisions in Article 430 have the potential to override the Code norms for general branch circuits as a result.
In particular, 430.6(A)(1) says that the amp rating from table 430.248 (for a single phase motor that is) shall be used instead of the nameplate amp rating when sizing the short-circuit protection and branch-circuit wiring, while 430.22 governs the branch-circuit ampacity for a dedicated branch circuit and 430.24 provides the ampacity calculations for a shared branch circuit. At the panel, 430.52 governs the maximum circuit breaker rating for a dedicated motor branch circuit for most folks, and 430.53 governs the circuit breaker for a shared branch circuit.
Your motor is even more special in that it lacks a built-in overload protector to keep the motor from crispy-frying itself if it seizes up or otherwise can't start/gets too bogged down. As a result, we need to apply 430.32(A)(1) and make sure that a 15A overload protection device is somewhere in the mix here. Having it share a branch circuit invokes 430.42(B) regarding overload protection and breaker sizing, too, while in the case of a dedicated branch circuit, 430.55 kicks in allowing an appropriately sized (as per 430.32) breaker to provide all protection functions.
Finally, we turn to 430.82 and 430.83(A) for the requirements for the switch that turns the motor on and off under normal conditions (the motor controller), while 430.101 through 430.112 provide the requirements for the switch used to turn the motor off for maintenance (the disconnecting means).
In your case...
Your motor is considered a 12A motor for branch-circuit sizing, which means that 14AWG wire is adequate for a dedicated branch circuit, but with a breaker anywhere from 15A to 30A in size (the minimum is set by 430.32, and if a dedicated overload protector is present, it can be increased to the 30A maximum set by 430.52). Likewise, the 13.8A nameplate full load amperes rating means that the overload device should be a 15A type.
Your motor controller can be a plain 15A/240VAC snap switch (lightswitch) as per 430.81(C)(2); it does not have to be double pole unless it is also the disconnecting means as per the exception to 430.84, as well as 430.109(C)(2) and the applicable portions of 430.111. If the controller is single pole, the disconnecting means can be the branch circuit disconnecting means for a dedicated branch circuit, or may be a cord-and-plug if the plug and receptacle are horsepower rated for the voltage and horsepower in question (240VAC/2HP for your motor) and the controller is located downstream of the cord-and-plug connection (i.e. unplugging the thing de-energizes the controller). Otherwise, a circuit breaker or switch switching all poles needs to be used to provide the motor & controller combination with its own disconnecting means as per 430.101 through 430.112, as applicable.
Welders are even more special
Welders are another special creature under the NEC -- they are covered by Article 630, which thankfully is much shorter and sweeter than the lengthy Article 430. The conductor and overcurrent protection ratings are both computed from a welder rating called primary, or input, current, as well as the welder's maximum duty cycle. This is as per 630.11(A) for the wires and 630.12 for the breaker.
When you go to shop for a welder
Based on the (sadly quite limited) data I can find out about the "50A" 3-in-1 units I've seen sold online and by the borgs, it seems that the specified primary current is somewhere around 32-35A with a duty of 60%. This means the maximum overcurrent protection for the welder and conductors is 70A and the minimum to avoid nuisance tripping of the breaker is 40A; a 50A breaker is a good, readily available middle ground. The minimum circuit ampacity in this case is 30A, requiring 10AWG wire -- of course, fatter wire won't hurt a thing.
What this all means for you
This means that you have the option of slamming things onto one branch circuit for now, or putting in a subpanel now and breaking everything up into its own dedicated branch circuit. I personally prefer the subpanel as it simplifies the motor-circuit design considerably especially if a two pole switch is used for the motor controller and disconnecting means, allowing the motor to be cord-and-plug connected in a more straightforward way.
Plan A -- subpanel time
The simpler, and easier to ensure compliance for option here, is to deploy a subpanel and feed the range, motor, and (future) welder off of dedicated branch circuits in the subpanel. This allows you to have the subpanel motor circuit breaker do all the disconnecting, overload protection, and short-circuit protection duties (you'll need a 15A 2 pole breaker for this), leaving only the motor controller function to be performed by a single pole switch (although you can use a two pole switch instead if you want to have it also serve as a motor disconnecting means). It also allows you to have the welder on its own circuit instead of repurposing the motor receptacle, and provides room for future expansion later provided you size the subpanel generously slots-wise to begin with (there is no such thing as overkill in this regard).
And yes, a 240V only (no neutral bar) subpanel is acceptable for feeding 240V-only loads using two pole breakers. You'll need the existing 30A two pole breaker (or a new one) for the range load provided it can be fed without a neutral, a 15A two pole breaker for the motor circuit, and a 40A or 50A two pole breaker for the welder circuit. If you wish to run the motor as a dedicated circuit with the welder getting its own circuit as well, then 14AWG can be used for the motor and 10AWG for the welder circuit, while a shared circuit will need to use 10AWG wire.
Finally, the feeder breaker should be a 60A or 70A two pole unit with a 100A molded case switch for the sub-main disconnect. This is safe for 4AWG copper at 60degC (you can push out to 80A at 75degC), and provides sufficient headroom for both the range and the welder to run without risking a nuisance trip of the feeder breaker. Note that using a breaker instead of the molded case switch for the submain, especially if the breaker is matched to the feeder ampacity, could lead to coordination problems as the two breakers "race" to trip on a gross fault.
Plan B -- everything on one branch circuit
Right now, it is permissible to put everything you are talking about so far on the same branch circuit. The 30A breaker would need to be replaced with a 50A breaker for a 45A computed protection need as per 430.53. The motor would need to be wired to a 50A, 2HP/240VAC plug and mating wall receptacle with a dedicated 15A unit mount breaker inline with this all serving as the overload protector (fortunately, you can use this as a motor controller too, and even as a disconnecting means if it is a two-pole unit). You would also need to use a 50A receptacle for the range due to the circuit being a 50A circuit.
If I had to do it this way, I would make a custom wiring assembly that brought the motor's cord and plug into a unit mount breaker enclosure, with a single pole 15A breaker in it that provides motor control and overload protection functions and the motor wired to the load side of the breaker. This way, unplugging the motor and plugging in the welder is simple when the time comes.
The other downside to this is that if the range and welder are running at the same time, you stand a decent chance of tripping the breaker in the future. Enlarging the breaker here isn't possible, even though the wires would support it, because standard NEMA receptacles only go up to 60A (and even then, 60A receptacles and plugs are rare).