It depends on alot of things, but you are on the right track. Much seamless tubing is simply drawn down to cold work the whole tube after welding. This pretty much makes the weld disappear. The cold worked weld ends up with properties closer to wrought material rather than cast properties of a weld. If plain welded tubing was annealed after welding, then the weld will not rust worse than the other part. If you only need a little bit, why skimp? the labor etc is a lot more problem than the cost of tube.
regarding 304 vs 316... there isnt a huge amount of difference really, either one will fail in poor circumstances. Every sailor is hot on 316, but it's not a huge difference. What's more important is the surface finish
, passivation, electro polishing or polishing.
Regarding welding electrodes, I cant remember now for sure, but in the aircraft industry, but I think 309 is used alot for most the 300 series or 321 which has titanium in it.