One amendment: 20k means dollars per basis point, rather than dollars of notional. That is, enough notional that a 0.01% change in interest rates would cause a $20 000 change in the net present value of the instrument. Traders often think in these terms because it makes it much easier to compare across different instruments.
Also, if you tried to trade 20k dollars notional, dealers would tell you to get lost, because an amount that tiny isn't worth their time.
traded like this because it is a meaningful measure of risk regardless of tenor. also its the same whether you're doing bonds or interest rate swaps (or, rarer, options)
B: size? = how much do you want to buy or sell?
A: 20k = $20,000
B: 2.4456/2.4511 = bid/offer = price/rate that A can sell/buy at
A: tx