My second effort failed. :-( Apps were still missing. Frustrated I did more Web surfing and found a very simple solution to the problem at the following Website:
See its suggestion # XXXXX That is, I connected the phone to the site, clicked the Apps button that it suggests (not the one on the left - the one I had been looking at that showed all my apps - but the one on the top/middle). When I click it, it shows me all of my original apps, with boxes on the left. For some reason only some of these boxes had been checked, the apps that I had been able to recover. Voila, I check all the unchecked boxes, and then press the Apply button at the bottom, which starts the sync. This time all apps with all their information get synced to the iPhone. What joy!
So, if anyone asks the question of missing apps following a sync, a good initial question to them would be, "Have you checked the apps button at the top, middle of itunes to see whether all apps have had their side boxes checked." In this case, such a suggestion would have saved both of us much time. Maybe you assumed that I had already done so, but I think it still would have helped to ask the question. I hope this message is clear. In conclusion, while your answers failed to solve my problem, all my apps are back on the iPhone with all information on them, including the Keeper app.