Is it normal for my heat pump to ice up when it's around freezing outside?

Yeah, that was the defrost cycle you heard. The whooshing is the unit reversing itself to pump warm refrigerant thru the outdoor coil for a few minutes, and the clicking is ussually the reversing valve or the fan stopping and starting. Then it melts the frost off, you sometimes get a plume of steam or vapor off the top of the unit that looks alarming if you're not expecting it, and it flips back to heating. On a damp night right around freezing it might do this every hour or so, sometimes more. Frost right at or just below freezing with high humidity is basially the worst case for frosting, so what you're describing is textbook normal.

The way to tell normal from a problem: watch it over a day or two. If the frost mostly melts off between cycles and the coil isn't turning into a solid block of ice, you're fine. If you come out and the whole unit is encased in a chunk of ice that never clears, or the fan blades are hitting ice, or the house stops keeping up with the thermostat, then call for service becuase the defrost sensor or board might be faulty. Also make sure the unit is sitting high enough that meltwater can drain away, if it's sitting in a puddle that refreezes it'll build ice from the bottom up. Don't pick at the ice or chip at it, the coil fins are soft aluminum and bend if you look at them wrong. If it ever does ice up solid and you need it cleared, just let it run or pour lukewarm water over it, never hot.

The green stuff on the casing slats is just algae or mildew growing on the paint in the damp weather, same stuff that shows up on siding. It's on the outside of the cabinet, nowhere near the actual airflow path, so it's not choking anything. If it bugs you, a rinse with the garden hose and a soft brush in warmer weather will take it off. Do not pressure wash it, you'll jam water and debris into the coil. Your filters being clean and the area being clear is what actually matters for airflow, and you've got that covered, so I'd just let it do its thing.

One thing I'd add from my own first winter with one: check where the water is coming from around the unit. Mine sat under the eave and every drip off the roof edge landed basically on top of it, wich loaded the coil with extra moisture on damp nights and made the frosting noticeably worse than the same unit at a friend's house out in the open. I put a downspout extension in and it made a real difference. Also, the frost you see through the grille is only part of it, there can be a fair bit on the coil face you can't see from that angle, so don't judge the whole coil by one spot.

Eh, I'd hold off on the hose advice for now. Mine grew the same green film on the cabinet and the reply says to rinse it off in warmer weather, but the guy is dealing with freezing temps right now. Any water you spray on that cabinet in this weather is just going to freeze on contact or run down and refreeze around the base of the unit. Wait till spring, the mildew isn't going anywhere in the meantime and it genuinely doesn't affect anything. Winter cleanup is how you end up with a new problem on top of the old one.

The downspout thing is real. Mine used to dump a sheet of meltwater right off the shingle edge onto the top of the unit every thaw, and it would frost up twice as fast as the neighbor's identical setup. I moved the outlet with a flexible extension and the difference showed up within a week, way less white on the coil in the mornings. Also second the bit about the base draining, I set mine on a couple of patio slabs when I installed it and the meltwater runs off instead of pooling and refreezing around the feet.