In the following sentence, should it have been "...father's telling me..."?

"Within five minutes of Father telling me that I should have to leave school I was wondering about the new suit I should wear to go to work in."

Strict grammar rules prefer 'apostrophe/ess', but either way seems OK to me.

No, I do not agree with father's. The only case when an apostrophe is equivalent to is is with it, it's. At all other places, it denotes possession.