Is it safe to screw a storm door right into hardie trim or will it crack?

Predrill everything and you'll be fine. That's the whole secret with this stuff. Use a masonry or carbide bit, go a hair smaller than the screw shank, and keep the drill speed moderate. If you hammer the trigger at full speed the friction heat can crack the surface, expecially near an edge. A little tape on the bit as a depth marker helps so you don't punch through the back of a thinner trim board by accident.

The trim boards are diferent from the plank siding. Trim is thicker and denser, so it actually holds a screw better than the lap stuff does. But you're right to worry about the hinge side. Being close to the edge is where chips happen. Two things help there. First, keep your pilot hole centered as best you can, at least a screw diameter away from the edge. Second, countersink very lightly or not at all. Most chip-outs come from the screw head seating too aggressively, not from the pilot hole itself. Snug the screws down and stop. This is fiber cement, it doesn't compress like wood, so cranking harder gets you nothing except a cracked or stripped hole.

Use the screws that came with the door if they're long enough to bite well into the trim. If not, corrosion resistant screws with a coarse thread work fine. Do not use drywall screws, they snap and they rust. As for the holes not chewing out over the years, a couple habits go a long way. Check the hinge screws once a season for the first year and re-snug any that loosened. If one ever does strip out, don't just keep driving it. Pull the screw, pack the hole with a two part epoxy or a plastic anchor made for cement board, and run the screw back in once it's set. That's what killed your old install, stripped holes getting torched down again and again til the whole face let go.

Last thing, the door sagging is often as much about the adjustability of the hinges as the screws. If the Andersen has adjustable hinges, get the door plumb with those before you final tighten anything.

Did mine last spring and the thing that made the biggest difference was sneaking the hinge screws into the wood framing behind the trim wherever I could. If your trim is thick enought, use screws long enough to reach past it into the jamb and you're not asking the cement board to carry the door at all. Grabbed a stud finder and was surprised how much solid wood was back there. Also put a shim under the bottom hinge rail before tightening anything so the door wasn't resting its whole weight on the screws. My last one sagged the same way and that was half the problem.

Chiming in to back up the framing point because it's exactly what saved my install. I drilled the holes in the trim a touch oversized so the threads never relly grabbed the fiber cement, then used longer screws so the bite came entirely from the jamb wood behind. Three winters in and the door hasn't lost a screw, no chips either. If your trim sits over solid framing on the hinge side that's the whole ballgame, the cement stuff just kind of goes along for the ride.

Update: I checked the trim thickness and the hinge screws can actually reach the jamb behind it, so I'm going the oversized hole route like you guys suggested. One thing though, the latch side and closer brackets sit where there isn't framing behind, just air. Do those screws biting into the cement alone hold up okay over time, or is there a trick for spots like that?