Wiring – replace a 15A breaker with a 20A breaker

The calculator you referenced is only appropriate for voltage drop and makes the disclaimer that the results of the calculation do not account for current specs or NEC. Any circuit fused for 30 amps must use a minimum of 10 ga copper or 8 ga alu. Longer runs may require an upgrade of wire size. In your case, use at least 10 copper for your welder regardless how far it is from the breaker panel. I'm sure one of my buddies here will have a NEC ref for ya.

Yes it is sufficient.

A 15A receptacle is designed such that an appliance with a 20A plug should not be able to plug into it. This means that at most under normal operating conditions of a 15A appliance, that the receptacle itself will only draw 15A. If the circuit shorts then the breaker will kick at 20A however.

The tab of a 15A receptacle is designed to handle a 20A load.

EDIT for clarification above: A duplex 15A receptacle contains a tab that connects the hot lead to the hot load. This allows the receptacle to carry the load of another outlet by wiring the lead of Outlet(2) to the load of Receptacle(1). When you break off this hot tab then you are specifically wiring for a switch to control only of the receptacles of the outlet, or you are specifically wiring for each receptacle to have its own circuit on a shared neutral. For two circuits to be wired to a single receptacle on a shared neutral, then you need to ensure that the breakers are on different service legs or you risk overloading the neutral. If you also break the neutral tab then you are wiring the receptacle to have two separate circuits each with its own dedicated neutral. These should be considered fringe cases for residential so it is probably best not to break the tabs if not needed so as to avoid confusing future DIY'ers and licensed electricians. Every time a DIY'er or electrician gets confused then they are potentially risking themselves.

The NEC states that this is acceptable as long as:

1. The wire is 12 AWG sized for a 20A circuit

2. There is more than one receptacle on the circuit.

It is important to note that under no circumstances should you install a single 15A receptacle onto a dedicated 20A circuit. This is against code.

210.21(B) Receptacles (1) Single Receptacle on an Individual Branch Circuit. A single receptacle installed on an individual branch circuit shall have an ampere rating not less than that of the branch circuit.

(3) Receptacle Ratings. Where connected to a branch circuit supplying two or more receptacles or outlets, receptacle ratings shall conform to the values listed in Table 210.21(B)(3), or where larger than 50 amperes, the receptacle rating shall not be less than the branch-circuit rating.

Table:

Circuit Rating (Amperes)
Receptacle Rating
(Amperes)
15 Not over 15
20 15 or 20
30 30
40 40 or 50
50 50

In conclusion, a standard 15A receptacle that one would buy in your hardware store is quality tested and approved to comply with all NEC codes to be legally sold in the United States. If the tab couldn't handle a 20A load without overheating then they would call this out specifically.