"afford" normally just refers to money, or something that there is a limited amount of. I can't afford to buy that Porsche. I can't afford the price of the opera tickets. I can't afford to waste any more time on this. I can't afford to gain any more weight. "manage" is more general. I have too much work and I can't manage it. I managed to come to work although I felt very sick. My new boss will be able to manage the office very well. The company managed to move my furniture in less than one hour.

No. If you "cannot manage this," it normally means that you cannot do something but the reason is completely unspecified. In other words, you may be "unable to manage to do something," because you don't have the ability to do it (either mentally or physically), because you don't have the time to do it, or you don't have the money to do it etc, or any number of other reasons. We use "afford," with time and money. "I cannot afford to waste time on this" = I have not got the time to do this. "I cannot afford this," I don't have enough money to pay for this.

In my mind, I equate "manage" in English with "riuscire" in italian. To use David's example: I managed to come to work although I felt very sick. Sono riuscito ad andare al lavoro anche se mi sono sentito male (I am not 100% sure of my Italian here :))

I have heard manage used as a synonym for afford but I wouldn't recommend it. It's rather colloquial.