I think the rules for dying are fine. DC 10 may sound easy, but for a typical person, that's a coin toss. They're as likely to waste their turn and accomplish nothing as they are to stabilize their friend. The person who is administering first aid is taking a turn out of the battle that could have been used to deal damage or contribute to victory in some other way. I've also found the death saving throws to be much less forgiving than 3e's bleed for 1 hp per round rule.
The only problem I have is with spare the dying. It's too easy to create a death yo-yo where characters pop up to life, get knocked back down, and pop back to life again. I think the best solution is to simply change that cantrip so that it only automatically stabilizes the target rather than healing them 1 hp.
I haven't seen any issues with the death and dying rules myself (I've not had any deaths, but I've seen a lot more *very narrowly* avoided deaths than in 3e or 4e. It definitely promotes drama), however if I was seeing a problem with this I'd probably say that repeat (successful) castings of Spare the Dying on a single target add a cumulative death saving throw penalty. So you're fine the first time it gets you up. The second time it gets you up, if you're dropped to 0 hit points you start on one death saving throw failed. Thus you die instantly if you're dropped to 0hp the fifth time (assuming StD is what gets you up each of the previous four times).
For my games, though, I love the way the dying rules actually allow me to have the monsters act more believably.
In 3e (and to a lesser extent 4e), having a monster strike a downed PC is basically instant death for them. Since my groups don't go for high churn, we've all avoided making monsters do that.
In 5e, if the monster *would* attack a downed character, I'll do it. The PC can take a hit, and instead of it meaning the PC just dies, the rest of the party know that they need to do something about the situation *immediately*. It's a general trend in the 5e rules that I like a lot: There's a lot of relaxation of the rules in ways that facilitate playing a role. I had some fun the weekend before last with two members of the party on 0hp inside a stinking cloud. Every able member of the party immediately went to gank the caster so that the cloud would dissipate before it killed the party cleric and fighter. Made for some nice combat *and* nice motivation.