Why other apple ID-s appearing constantly?
It sounds like you are restoring the phone from a backup (iCloud or otherwise) and such a backup is placing all of your old apps back on your phone, which it is supposed to do.

It sounds like what is happening that some of the apps were downloaded under another AppleID. Whether it is someone who was using your phone and did this or the apps are from other sources.

The only way to get around this (without jailbreaking your phone) is to cancel all app installs and then download them one at a time from the app store under your personal AppleID. Apps that are associated with your current AppleID will have the download icon next to them (cloud with a downward pointing arrow) all others will have have a "GET" or "BUY" button.