Q1: If you are connecting to an Exchange server, and I think also if you are using the Hotmail Connector, you may see synchronisation logs. Essentially this log will list any issues it may encounter when synchronising to these types of servers. I used to get them when I was connected to an Exchange server and for the most part they are usually one off issues such as your computer ran into performance problems which caused an issue with synchronisation. If you are concerned about any in particular just google the error message and you should be able to find some possible solutions to the problems.

Q2: An .ost file is still a Personal Folders file in Outlook however this file type is used when connecting to an Exchange server or using the Hotmail connector The .ost provides offline access to these account types. Judging by these two questions having a link I am thinking you are definetley using either an Exchange server connection or Hotmail connector.

I hope this helps and let me know if you have any further questions.