The carry is the PNL resulting from holding a position. However, even if you don't finance the bond in repo, you can still measure your carry as the yield-to-maturity of maturity of the bond vs the yield of the alternative default investment you would have made with your cash (for example 0% if sitting on your bank account at 0%, but maybe it'd be 1%, etc).
The formula you mention [carry = fwd yield - spot yield] is due to the arbitrage-free assumption: say carry > fwd yield - spot yield, then the fwd yield is priced too low and I could sell the bond fwd, buy it spot, hold it until the fwd delivery date and make a positive PNL.
Roll down is the mark-to-market due to the passage of time assuming that the shape of the curve doesn't change. This is a strong assumption and has a few limitations.
In this post I discuss what carry and roll are, and look at the bond future's asset swap as well: http://swapsball.net/how-to-calculate-carry-and-roll-down-for-a-bond-futures-asset-swap/