How do I know if a Hello Fresh shipment is ... fresh?

The short answer is you don't know, there's no certainty. Even with food bought from a supermarket there are no guarantees, there could be handling errors which reduce the shelf life or you could get it home and find it's spoiled.

With your package if the food was still very cold to the touch and the ice pack was still frozen then there's a good chance the food is still fine, however a cold pack can feel chilly but not be cold enough so unless you measured the temperature you can't be sure.

I would treat the meat with suspicion, and play it safe. If you think it was kept refrigerator temperature the whole time and the meat looks and smells fresh then it may be safe, if not I'd chuck the meats as it's not worth the risk. Cheeses may be fine, many get aged above refrigerator temperature, so if they didn't stay that cold they're a better bet.

Pretty much the same as you would do with the regular supermarket-bought food: check for the package bloating, odd smells, discoloration, sliminess to the touch, all the usuals. It would also make sense to slow cook the meats. Given the snowstorms and unmelted ice, it is likely the foods have spent their time on a truck in favorable conditions.

Personally, I would cook it unless given an indication something's off (like odor), but that depends on how sure you are in that it is safe to eat. When in doubt, throw it out.

HelloFresh boxes can only be refrigerated, not frozen after they are packed, because freezing would ruin the lettuce, spinach, etc. Therefore if the ice in the meat pack is still frozen, then your box was almost certainly packed today or yesterday.

They probably packed your box 4 days late knowing that they wouldn't be able to send the driver into the storm. Your food is fresh.

On rare occasions HelloFresh sends me stuff that is less-than-fresh; if it looks or smells gross, I don't eat it. And I eat the fish and chicken first because I have noticed that when they come from HelloFresh they do sometimes expire a couple days before the expiry date printed on the packet.

The package was cold, ice pack frozen, etc. when you received it. But the big question is whether it stayed cold the whole time. There are temperature monitoring labels such as this one:

(I have never used this particular one, just found it in a search, there are many manufacturers of similar items.)

But residential shipments don't necessarily have such things.

The other useful clue is the "ice pack".

If it contains dry ice (solid CO2), it will sublimate over time and the packs will be partly or fully empty if the temperature has risen too high for too long. But again, residential deliveries are probably less likely to use dry ice due to handling concerns.

If it contains water ice but is well sealed, it will melt if the temperature has risen too high for too long, but it will refreeze if the temperature is cold enough for long enough. If the pack normally has "cubes", a refreeze will be obvious because the pack will feel differently (single blob instead of "cubes").

If it contains some other liquid/solid, all bets are off. Many "ice packs" contain a liquid that can be frozen/melted/refrozen numerous times with no obvious effects.