The answer to the questions are;
1. No.
If you were pilot in command then you can log Pilot in command time. If you were pilot in command then you did not need a check beause you were not getting one.
If you were pilot in command then anyone else in the aircraft with you was a passenger and you must have been entitled legally and under your club rules and insurance to be pilot in command of an aircraft carring passengers...........so no checkout was required........so you did not do a checkout.
If you were not so entitled then you were operating either illegally, in breach of the club rules or in breach of the insurance requirement (thus illegal) or any combination of all three.
2. No
You can not change the entries in your logbook ans that would be falsifying the entry. If you think that there is an error then before changing the entry from dual to Pilot in command, you need to contact the person who is named as pilot in command and ensure that they agree that there was an error and that they have not also logged the time as pilot in command.
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It has nothing to do with who wants to log the hours, it is a legal requirement for the pilot in command to log the time as such and for the other pilot not to log pilot in command time.
The CAA can and do check logbooks and can check entries are correct. The fine for false entries i.e. one such as claiming to be pilot in command when one was not and the aircraft / club / instructor records say you were not is about £2500 per line.
Even at £400 per hour there is a few hours in that money.
Regards,
DFC