I am 19, and apparently I had a cached belief that 16 is the ideal age for childbirth. (I've tried to track down the source, and I think it's from a novel I read a really, really long time ago, where a character was 'legally too young, but biologically the ideal age for childbirth'.) A quick Google search suggested 25-35 years of age as the period of peak fertility. Which I did not know. And which makes me feel better about having to delay having kids until then.
On the main topic, there's a big danger of generalising from one example: whether you find babies cute is likely to relate to a whole host of your personal experiences and feelings about babies as well as the instinctive cuteness response.
No doubt. But in general, I think a LOT of people (especially females) will have had the personal experiences that lead them to think babies are cute. And I wouldn't be surprised if mothers whose 'cuteness' instinctive response is lower would have more trouble raising children, no matter how good their intentions. (I have a very sad story about this, actually, but I'm saving it for a top-level post.)
If I was making up just-so stories, I'd guess that cuteness serves to get attention, increase patience and prevent boredom during childcare, rather than to make us want to look after them.
This reminds me of the area of qualitative research (in nursing, but you can do it anywhere I assume.) You go out and interview a whole bunch of people (mothers with babies in this case) and ask them a lot of questions about the emotions they feel surrounding their child and how their warm fuzzy feelings affect the way they care for your child. Then you compile the results, pick out common trends, and you have some empirical evidence to justify your just-so stories. (Assuming that baby-cuteness serves the same purpose now as it did during our evolution, which I think is safe.)
As an aside, I really don't have much of a cuteness response to animals. I occasionally feel guilty eating meat because of a top-down moral belief that they have some form of consciousness and ability to feel pain, but on a purely emotional level I doubt I would have any trouble killing and eating a rabbit.
If you're not running on instincts then you might want to be particularly careful with your beliefs in this area...
Peak fertility is different that the optimal age for a first child. Fertility is much easier to measure (based simply on the probability of getting pregnant given an standard opportunity to do so) whereas the best age to have your first child is a ridiculously complicated calculation having to do with your values and goals plus: the current and future state of medicine, the current and future state of the economy, your current and future pool...
Daniel Dennett has advanced the opinion that the evolutionary purpose of the cuteness response in humans is to make us respond positively to babies. This does seem plausible. Babies are pretty cute, after all. It's a tempting explanation.
Here is one of the cutest baby pictures I found on a Google search.
And this is a bunny.
Correct me if I'm wrong, but the bunny is about 75,119 times cuter than the baby.
Now, bunnies are not evolutionarily important for humans to like and want to nurture. In fact, bunnies are edible. By rights, my evolutionary response to the bunny should be "mmm, needs a sprig of rosemary and thirty minutes on a spit". But instead, that bunny - and not the baby or any other baby I've seen - strikes the epicenter of my cuteness response, and being more baby-like along any dimension would not improve the bunny. It would not look better bald. It would not be improved with little round humanlike ears. It would not be more precious with thumbs, easier to love if it had no tail, more adorable if it were enlarged to weigh about seven pounds.
If "awwww" is a response designed to make me love human babies and everything else that makes me go "awwww" is a mere side effect of that engineered reaction, it is drastically misaimed. Other responses for which we have similar evolutionary psychology explanations don't seem badly targeted in this way. If they miss their supposed objects at all, at least it's not in most people. (Furries, for instance, exist, but they're not a common variation on human sexual interest - the most generally applicable superstimuli for sexiness look like at-least-superficially healthy, mature humans with prominent human sexual characteristics.) We've invested enough energy into transforming our food landscape that we can happily eat virtual poison, but that's a departure from the ancestral environment - bunnies? All natural, every whisker.1
It is embarrassingly easy to come up with evolutionary psychology stories to explain little segments of data and have it sound good to a surface understanding of how evolution works. Why are babies cute? They have to be, so we'll take care of them. And then someone with a slightly better cause and effect understanding turns it right-side-up, as Dennett has, and then it sounds really clever. You can have this entire conversation without mentioning bunnies (or kittens or jerboas or any other adorable thing). But by excluding those items from a discussion that is, ostensibly, about cuteness, you do not have a hypothesis that actually fits all of the data - only the data that seems relevant to the answer that presents itself immediately.
Evo-psych explanations are tempting even when they're cheaply wrong, because the knowledge you need to construct ones that sound good to the educated is itself not cheap at all. You have to know lots of stuff about what "motivates" evolutionary changes, reject group selection, understand that the brain is just an organ, dispel the illusion of little XML tags attached to objects in the world calling them "cute" or "pretty" or anything else - but you also have to account for a decent proportion of the facts to not be steering completely left of reality.
Humans are frickin' complicated beasties. It's a hard, hard job to model us in a way that says anything useful without contradicting information we have about ourselves. But that's no excuse for abandoning the task. What causes the cuteness response? Why is that bunny so outrageously adorable? Why are babies, well, pretty cute? I don't know - but I'm pretty sure it's not the cheap reason, because evolution doesn't want me to nurture bunnies. Inasmuch as it wants me to react to bunnies, it wants me to eat them, or at least be motivated to keep them away from my salad fixings.
1It is possible that the bunny depicted is a domestic specimen, but it doesn't look like it to me. In any event, I chose it for being a really great example; there are many decidedly wild animals that are also cuter than cute human babies.