1. No
2. Yes
You're right, if the figures you have assumed are correct: the amount of lift for the given scenario would be the same at altitude as at msl.
However, this given amount of lift would have to be density-compensated (as you correctly state) by an increase in thrust (Increasing T req, decreasing T av) and decrease in Alpha accomplished by a lower pitch attitude.
The next problem is whether climbing at the constant IAS compromises the stalling speed of the aircraft with decreasing density (it will do eventually), so the transition to constant Mach No. has to occur.
If anyone has other ideas/opinions/corrections I'd be glad to hear.
Charlienov.