I do lots of tires it seems....

For me it is a matter of getting it done NOW, or spending the time for someone else to do it.
My 2 cents, go to Napa buy a bottle of tire soap. It is about $5 and will last you a long, long time.
My next question will be do you have it clamped down somehow? Chasing those things around is no fun.
I have a couple different tools and clamp's to do the various tires.
As to stretching it over the rim, first thing is, do you have the right size tire?
If you do, then I bet the opposite side of the bead is not down in the recess as has been mentioned here before.
As to pinching the tubes. Tire irons, well polished and smooth go a long way, but I had something pointed out that really helped me with that problem.
If you do not push your tire iron past 90 degrees, you will not pinch the tube. You will have to take more bites, but the tube will not be pinched.
When I was 17 or so I worked on a tire service truck for a while and was paired with this old (probably in his 70's) black guy that probably went 100 lbs soaking wet with change in his pockets.
I would flail and flail at the tires with the tire hammer trying to get the beads to break, guess you had to be there but it was almost comical the speed that he would walk over too me and go, Now Alan,,,,,,, just hit it right, not a lot...............
I wish I could explain the picture better, but he would take that hammer from me, stand up on the tire, slide the hammer around behind him, then it would kind of pivot off his shoulder and more "fall" onto the tire then anything, and I kid you not a bit, the tire would fall off the bead.
He was also the guy though that would fall asleep airing up a split ring tire with his hand holding the chuck on the tire. Would wake up, check the air pressure, and it would be right................. Always scared me to death when he did that.
Anyway, if you can find someone experienced to walk you through it the first time, it is a good skill to have, and so much easier to learn by "seeing" then by reading.
Good luck, hope you get it fixed.