You will not need your logbook during airline training.

Many instructors will not sign your book even if you ask them to... they don't want the liability if you use their training for GA currency. Also not all of them are CFIs.

I wouldn't even log that SIM time, but if you must you'll need to keep copies of the airline records, since SIM requires an instructor signature to be legal.

For IOE you log that yourself. It is not logged as dual received, it's logged as SIC, so again no instructor signature.