Can departmental funding combine with fellowships/grants?
In general, the rule of funding in graduate programs is that external money is used to replace internal money. That is, if your department gives a stipend of, say, $20,000 per year, and you win a fellowship that pays $15,000 per year, the $15,000 will be used to offset part of the $20,000. 

However, most departments that I'm aware of will also guarantee that fellowship and grant holders don't suffer relative to their peers by bringing in outside money. That is, they guarantee to make up any shortfall between the value of an external award and the department's standard. (In the sciences and engineering, I would steer clear of any department that doesn't have such a policy!) Many departments will even incentivize bringing in outside award fellowships by giving a premium or bonus that supplements the outside award. This might be done either by multiplying the standard award by a percentage (so, for instance your total would be 125% of the department's standard), or by giving a lump sum on top of an outside award (if it exceeds the standard, for instance). 

One thing to watch out for, however, are restrictions placed on the outside fellowship: your department may have to certify, for instance, that you will not be assigned teaching duties or other non-research duties during the fellowship period. If you are unable to attain such a certification, the award may not be given out. 

For the PhD phase of a program, the tuition should be covered by the department and the outside fellowship; however, in general the student does not receive part of the tuition payment back. In general, the tuition charge is a negotiation between the university and the fellowship program, as this is at the graduate level usually a financial transaction. Also, if the student were to receive the tuition payment which was then forwarded to the school, this could be called taxable income, which would make things very messy for everyone.