Your dryer outlet relies on the neutral and ground being connected somewhere, somehow. More recently apliance plugs & outlets are often 4 prong, 2 hots a neutral and a ground. Modern electrical pannels don't connect the ground and neutral busses, EACH BUSS has a ground lead to a rod driven in the ground. You DO NEED a doubble pole breaker if You add a circut. The 16 amps listed on the welder requires a minimum 20 amp feed, that is #12 wire unless unusually long. Your welder needs 2 conductors with ground. There is nothing wrong with running a 3 conductor with ground wire, that enables You to tap out 115 volt if You need it at a later time. #10 wire and a 30 amp breaker is not a problem other than higher cost, and gives flexability for the future. You could hard wire the welder cord right to the breaker, or You could use a 30 amp 3 or 4 prong recepticle and plug. I used a dryer cord on My old farm welder. You can get a recepticle and plug from a welding suply store, a farm store or a good hardware store. Home Depot & Lowes should have the recepticle & apliance cord, maybee the plug. On the welder the green wire is the ground, the other 2 are hot.

Double check/Triple verify what voltage is at the receptacle. If the machine has an input linkage for specific primary voltages; the machine may not work very good. ie if the house voltage is 208 and the input linkage is 240 the machine will weld "colder". Or if the machine is linked for 208 and the receptacle is 240 the machine will weld "hotter".