> Lastly bond prices have an inverse relationship with interest rates, which means that as interest rates rise, bond prices go down. So no, when rates go up to x+5 you will never be able to find bonds that pay x+5.

Indeed, interest rates are inversely related to bond prices. And bond prices are inversely related to yields. Yields and rates are highly correlated. In fact, the way banks finance mortgages is by selling bonds. The market rate they can get on those bonds determines the rate they can offer to homeowners. So of course these rates move in tandem.