Q:

What is the word for a person who does different jobs?

I'm asking about someone who's employed by a company and does different jobs (both a qualified and unqualified person). Who knows; maybe in English those two are two different words? What are they called in English?
All I know is a navvy but that usually means physical work.
Let me just try a few things here. These are just wild guesses, nothing more. 

odd job man/worker  
shiftman  
(unskilled) labourer

A:

A person who can perform many jobs may be a:
Jack of all trades

A:

I am not big on jack of all trades used in a "company" sense.  Not many people would say, let us get our jack of all trades to help you out. 
I have heard one specific title given to this type of person at restaurants, construction jobs, and big company.  The term is floater.  

One who wanders; a drifter. 3. An employee who is reassigned from job to job or shift to shift within an operation.

A:

For formally communicating a person's role in the company, I would consider referring to him or her as a generalist. Contrast this to a specialist, whose job is defined very narrowly.
Our generalist, Samantha, is responsible for performing a wide variety of tasks to ensure we're shipping the best product possible.
Definitions: Merriam-Webster,  Oxford.
It's difficult to imagine unskilled laborers being referred to as generalists and specialists, though, since being unskilled precludes one from having specialized in something.

