A job title on a resume helps you get a job by showing you have already held the same job, or a job that's considered a prerequisite, as the job you're trying to get. You're not trying to get a job, so this is irrelevant.
A job title on fb/linkedin can serve the same purpose as a resume, or can be simply for showing off to friends and family that you are now an Intermediate Assistant Analyst rather than the Junior Assistant Analyst you once were.
A job title on a business card lets people you're dealing with outside your company know what your job is and what they should and shouldn't discuss with you. Same holds for an email signature.
The first case doesn't apply (you believe you may work in the family firm indefinitely) and the second is offtopic here (tell your high school friends you're the VP if you like, whatever.) So that leaves the third item.
I laughed aloud when you said you did "everything" except, you know, what the firm actually does for money - scaffolding, selling scaffolding jobs, and planning how to get the jobs done with the people and equipment you have. Those three things are "everything" - what you're doing is important, but it's nowhere near everything. If I read "Finance and Administration Manager" I would expect the person decided whether the company needed loans, or to buy new equipment, as well as setting revenue goals and such. I think that's a little over your head. "Office Manager" wouldn't be bad. Or "Finance and Administration Clerk", though Clerk is pretty old fashioned these days. You could try for something modern, funny, and hip, like "Head Paper Herder" or "Chaos Master" if you think your customers would get it.
Or you could not worry about it. Especially if your last name is the same as your dad's, people who exchange emails are likely to know your role. You're pitching in and doing what needs to be done, and your father trusts you with millions of dollars. That's more valuable than any title you can make up.