Ah yes, this is a somewhat popular rates/equity hybrid structured note called “contingent range accruing” notes or something like that, depending on the issuer.
“So, if I’m right, if the two year rate is 3% and the 30-year rate is 5%, then it will pay 7 x 2%, or 14%, or 10%, whichever is less.”
Correct.
“1.) What is the Constant Maturity Swap rate? Where can I find the rate curve? What might cause the spread to narrow?”
CMS is pretty simple to understand. For example, the 30y CMS rate is the 30y rate, from the observation date. So if today is 8/22/2017, the 30y CMS rate is the swap rate for 8/22/2047. Tomorrow, the 30y CMS rate will be the rate for 8/23/2047. This is counter to conventional rates/swaps pricing, which is based on fixed dates, not fixed terms from the observation date. For instance, under conventional (non constant maturity) rates trades, you will get payments based on LIBOR or another rate on a fixed date, say 8/22/2018, while the constant maturity rate rolls forward based on what day it is today.
You can derive the CMS from the normal yield curve. For instance, if you want to know the market estimate of the 30y CMS rate 1y from today, you would find the 1y/31y forward rate from the yield curve - basic CFA material.
In this case, the spread narrowing means the yield curve will have become less steep. This can happen due to a variety of macroeconomic effects.
“2.) Why would anybody issue this kind of bond?”
The same reason they would issue any kind of bond - to charge a spread. The seller of this bond can hedge all market risks. For instance, the issuer is short some kind of call option on SPX, since the coupons are paid only if SPX is “850 up” in 5 years. So, if the issuer can buy this option in the market for a lower implied volatility, they can hedge their exposure while collecting a spread. Similarly, the issuer is long interest rate volatility, short interest rate term structure, and short rates/equity implied correlation. So, their goal is to hedge these risks at levels better than those offered in the structured note.