Will Google play store auto update in future if I install latest store APK manually for now? I'm desperately waiting for play store to update to 4.9.13, now it's been three weeks since update release and I'm losing control now. I want to ask that for now if I update play store manually with the latest APK from android police website, will I receive next update as an OTA or will I have to update it manually? And will there be any problem if I install play store update manually with APK?

Thanks in advance!
Yes, It will :) 

I do this all the time when a update comes up and available to my phone. 

I too am using Play Store 4.9.13. 

Even if this doesn't work out, you can uninstall updates in app settings in your phone.