Welcome to EBN!
All you describe IS typical bulldog behavior, they want to always be with their humans it is how they get the nick-name 'Velcro-dog'. All four of my babies have had some level of issue with a moving object... broom, vacuum cleaner, empty trash bag, dust mop... all typical. Heck, my Dyson has scars from Banks attacking it.
Beckett may also be looking for leadership, her high energy and anxious antics is most likely related to not knowing who is the leader in the household. Are you and your wife using all the same limitations and commands with her? Crating is not going to solve the issue, it may make it worse as she will she is as punishment/separation and cause more anxiety ... try taking her for a long walk or playing fetch in the yard... a tired puppy/dog is a good puppy/dog.
As for shedding, my guys barely shed unless it is the 2x a year they are blowing their coat... give her some vitamin D it should help with the shedding, but also keep in mind even though she is on a fantastic food (THK), it may not be the right one for her, maybe try a different protein?
Lastly, the issue may even lay in the one statement... 'overall, we really don't trust her'. Beckett senses and knows this which gives her an unbalanced feeling in the leadership of the household, which in turn causes her anxiety, humping and what appears to be un-acceptance of the children. You and your wife need to give her consistent structure and limitations, but not punishment.
Hope this helps.... please keep us posted