Start by looking forward to a very interesting, strange, and often funny experience!
Haven't posted in ages, but as someone who had their second at the 'advanced' age of 36 maybe I can offer some advice/comfort on the geriatric pregnancy thing. First off, what asshole came up with that terminology? Nothing makes you feel more ready to have a kid than being labeled 'geriatric'. But fear not. What we lack in energy we make up in dark, inappropriate humor, especially when the birthing part comes.
Health-wise, I won't lie, it was pretty draining. I don't know if it was age or it being the second, but I was wiped with #2. What did help, and might give you guys some confidence if you get to that point, are the things you can do to prep physically that made a big difference for me (and many others). Squats. Do squats. Do looooooots of squats. They don't just work the area that's going to take the brunt of the birth, they make the whole process easier and speed up recovery like no ones business. If you do nothing else exercise-wise, at the very least do squats. Kegels, sit-ups, stretching (yoga-type stuff), and keeping limber are also really great too. If your wife starts now, a good few months before the birth, I think you'll feel less nervous about the whole thing.
Tests-wise, yeah, it's pretty stressful all the things they throw at you. Best you can do is remind yourself that they're just going through their own checklists and the odds are still very, very small. We did the same as you and kept mum until we got the results back from our tests, which made us feel silly after for worrying so much (over the tests, not the keeping mum part). :P
Life-wise, it sounds like you're in a good place, so rejoice! One of the few advantages we have over younger parents to be is generally we're in slightly better position. While having a kid may feel bizarre after having everything to yourselves for so long, it's not that bad. Really. I had my first at 34, and I still get my free time, my alone time, I still play games, my husband still goes to the gym 4x a week and gets his private time, etc.*. Life doesn't end after a kid. It just adjusts. You guys'll figure it out.<--And that'll take time and poop. Don't think for a second any of us had a clue at first, so don't stress.
*YMMV in the first 6 months
Also, congrats to all the new parents! And best of luck,
eZipsis. You're not alone in the eating stress area.