The quick answer - No, not really, on any of the points..
Cost-of-carry (and its counterpart, roll-down) are concepts very misunderstood in the fixed income world if you ask me. Most people seem to have their own, ill-defined definitions of what cost-of-carry and roll-down are. Sometimes people interchange or even bastardise these terms.
My definition is as follows:
Cost-of-carry: the physically payable cost for holding a trade for a defined period of time. A bond purchased on repo would incur the cost of financing the purchase, for example. Regulatory capital charges would be another example for a bank.
Roll-down: the mark-to-market (MTM) gain or loss on a position if the yield curve remains static over a period of time. For example; today 9Y and 10Y yields might be, say, 1.870% and 1.917% and in 1Y time assume they will still be 1.870% and 1.917% (even though today's fwd curve does not price that outcome - it actually prices 1Y9Y at, say, 1.943%)
So if you buy a 10Y bond with an annual 3% coupon at a yield of 1.917% for a dirty price of 109.81, financed by a repo rate of say 1.70% what will happen?
Your cost-of-carry is the repo cost minus the coupon = (109.81*1.7%=1.87) - 3 = 1.13 gain.
Your mark-to-market is affected by the new yield of 1.943% and price of 108.68, meaning a loss of 1.13.
So if the yield curve evolves as expected you gain cash and lose on MTM in the same amounts. This is arbitrage free pricing in action. If this didn't happen everyone would purchase bonds financed on repo and hope to accrue gains.
But now lets assume roll-down: if the yield curve remains the same then the price of the bond is instead 109.10 so you only lose 0.71 on MTM and hence your net gain is 0.42.
These example numbers are all consistent within the context I created in a spreadsheet for a specific term structure, but in the real world different bonds can have different repos, the term structure of the curve will be different, but the principle are the same.
For your two specific bonds at 5% (long) and 2% (short), if they both have the same repo then after a year your cost-of-carry will be +3 whilst your MTM will be -3 under an arbitrage free pricing model. If you assume roll down then this will depend upon the shape of the curve it is entirely possible that depending on which curves you have purchased you will see a negative roll-down.
This other answer shares the same opinion.
Another answer I give to swaps shares the same concept.