Do the monsters die in this game? Or do they just have to be sealed as in the more recent games?

They don't die. Once they stop gaining stats and AP, you can seal them, or you can continue to use them in battles/performances/etc. In my experience, older monsters are more injury-prone, so you have to be careful with them.

Also, since most of the stats carry over when you combine, there's really no big reason to keep an old monster around when its child will be just as capable in most respects.

I've been holding on to older monsters when I have baby monsters to try who I want to get up in stats, do you think that this is useless, I should just make new ones right away??

Well, it's not useless. But say you have a monster in old age with 15 Pow/10 Acc, big enough to take on most of the adventures post-end boss and shepherd new babies along to get some needed AP.

You could keep the oldster alive and get the babies AP, but he won't be getting any new stats or AP as you train him and do shows. So, you could convert the oldster into a saucer, and mix him with some other saucer, and you'll get a young monster with probably the same or very close to 15 Pow/10 Acc, still powerful enough to shepherd your first gen babies, but he'll be able to benefit from the training, and he'll get more powerful.

So if you're talking just holding him around for an extra month or two, that's not so bad. But if you're thinking of having him hang around for more time, might as well saucer/combine him to get something new that can grow.