Is It Safe to Inject Hyaluronidase on Several Areas?

It is possible to have a reaction to Hyaluronidase, but the side effects are rare. Side effects include redness and pain at the injection site or a severe allergic reaction.
We agree with Dr. Chen that you should pick the more pronounced areas to reverse. One of the main reasons for this is that we would expect the effects of the Juvederm to start diminishing very soon, which would remove the bumpiness naturally as the product wears off. No reason to inject Hyaluronidase in areas that have Juvederm that is not visible or barely visible.

It should be ok to get hyaluronidase in multiple areas in a controlled fashion. Alternatively you could get treatments over several days to stagger the treatment.

Your concerns are all valid. Juvederm is safe to inject in multiple areas. Yes, it could break down your natural collagen but injections of hyaluronidase do tend to stay localized when small volumes are injected.
In light of your concerns, I would select a couple of areas on less prominent areas of the face to inject small volumes of hyaluronidase and re-assess in 1 week.
I would encourage you to return to the practitioner who injected the product and express your concerns. We generally like to know when our patients encounter problems related to our work.
Good luck.