is it safe to use a microwave with the outside glass missing?

DANGER, WILL ROBINSON

The glass doesn't stop the microwaves. The mesh does.

However, without that glass in place, the chance of something else going wrong - e.g., that foil coming off or some other part coming loose, is drastically increased.

As far as replacing with plastic, I definitely don't recommend it. I'm not against plastic for certain repairs - e.g., quick fix for a broken taillight cover. But for a microwave oven door where the risks are really high if something goes wrong, absolutely not.

This falls into the "what you don't know can really hurt you" category. Unlike a conventional oven (just "heat", which you can feel as it melts the plastic you put in there, producing toxic fumes, etc.), with a microwave oven you could undergo serious bodily injury and not know it is happening until it is too late.

No. Just no.