MeTwo: knowledgeable people have done things far more dangerous than you're planning and survived it. That doesn't mean you'll be in that company though.
If you persist, it's better that you do so with advice than without. Consider this advice rather than ecouragement-
If all you're after is a short-term experiment, bottled gas is the way to go. Forget about procuring a compressor for a short-term experiment. And even if you go with a compressor, forget about getting any benefit from the work of compression at this tiny flow. Heat loss will eat it all.
As to your question, you need the intercooling because the compressor materials won't survive if they're run too hot. This question on your part has me wondering if you know enough about this stuff to survive this experiment.
You won't successfully operate a 1000 C electric furnace in "an open field", so you're being a bit disingenuous- you at very least need a roof over this thing.
You'll also need to have whatever is receiving this hot gas inside the same furnace. At these conditions, a few inches of tubing will lose you 100 C regardless how well you insulate. For your preheater, consider putting an ARI Firerod heat trace inside your tubing, with the air flowing in the annulus between the heat trace and the tubing ID. They won't last long heating air to 1000 C though!
The 309 or 310 SS tubing is nice in theory, but the trouble will be the hot end connection. Forget about Swagelok-type compression fitting connections- they're not available in 310SS, 316SS won't cut it for long, and that fitting design is not going to hold together at these conditions to the rupture strength of the tubing anyway.
Figure out what your hydrotest pressure would be at the ratio of allowable stresses between room temperature and 1000 C and you'll get an idea of what you're up against.
Bottom line: you need cone and thread connections or the like, or a very risky weld. Have a look at Autoclave Engineers.
You've given no indication of what's on the downstream end of this. No valve is going to survive throttling a small flow of extremely hot air- you'll need to cool first.