All of the mentioned processes will work with certain material status awareness and atmospheric condition controls. The older, somewhat less shiney material can best be done with stick. Stick is oblivious to air conditions and can handle rather rough material situations. "Good" material can be done with MIG and shielding gas if conditions allow. Near perfect material can be done with TIG and with minimal air disturbances prevailing.
The secret to a satisfactory frame project is in your set-up. Everything needs to be straight, parallel, perpendicular and well clamped before the welds are applied. Welding should be done in a manner that accomplishes the goal of maintaining all the proper alignments. This can require various tie-bars, braces, and many clamps along with suffcient supports.
I suggest E6010, E6011, or E7018 stick electrodes. 6010 and 7018 are usually DC Reverse electrodes but there are A/C 7018 rods marketed. My MIG wire suggestion would be ER70S6 with your favorite shielding gas. I like 75/25 but others work as well or better. I suggest using the TIG process only as a last resort for your frame work.