When I was looking into either renting or buying a cell phone to use in Spain, I found several sites that rent cell phones. I'm not sure how long you're staying in Madrid, but it may be worth it to just buy a pre-paid cell phone.
Personally, I actually ended up buying an unlocked cell phone and a prepaid Spanish sim card to use with it through Ebay. Renting didn't seem economical, especially since I didn't get to keep the phone and high shipping charges added to the cost. And, I bought my phone and sim card in advance of arriving in Spain because I wanted to know my telephone number before I left.
If you're calling home, the cheapest method is probably a phone card. That's what I did. If I had used my cell phone, I think it would have cost about 0.75 euros a minute to the US. As to having your family call you on a cell phone, keep in mind that, in Spain, the caller is the one who pays the costs of calling, so it's pretty pricey, especially if it's an international call on top of that. I had a cell phone so that my family could call me for an emergency, but I actually did all of the calling with a phone card.
If you decide that you want to buy a cell phone and if you want to be able to use it in the US afterwards, it needs to be an unlocked tri-band phone. Dual-band phones won't work in the US.
Also, if you're not in Spain yet, it should be pretty easy to buy a phone when you get there. When I was in Spain, I saw ads for pre-paid phone packages in the window of the Telefonica resellers ...A cell phone and $70 to $100 in saldo for $80-$100 or somewhere around there. This is probably what your friend thought was a "disposible" phone. The phone itself doesn't expire. It's the SIM card that expires if you don't recharge it by adding more $ to it before the expiration date.
As for what company to pick, my friend used Telefonica and was very happy with it. I actually used Vodafone and I thought it was good. So, I think any of the big companies would work...I can't help you out with per minute charges as I don't know what mine were....I just figured I'd use my phone until I ran out of saldo. I actually didn't end up having to recharge during my trip.
Hope this helps!