New stem, bars & shifters on old Bridgestone 700
you should be able to get a variety of 22.2mm quill stems. For instance SJS have a fair selection (picked because they came up first in a search for 22.2mm quill stem, I have no connection with them). 22.2 is a standard size and still common.
If it has indexed shifters you should have no problems, but Shimano 600 may not be indexed, and might be 6 speed. In which case you can look for thumbshifters with a friction mode or change the drivetrain. The cost of buying new 7 speed drivetrain parts is enough that I wouldn't suggest it - you'll spend a fair chunk of the cost a new cheap bike doing it. If there's a bike recycling group or tip shop in your area you should be able to avoid spending most of the money - old bikes with 7 speed indexed gears are pretty common.