Bike or e-bike/trike good for the rain
Short Answer

I believe the short answer to your question is no. The first three requirements that you list put you in the 'deeply customized' bicycle neighborhood. This is in direct opposition to your fourth requirement--custom bikes are very expensive. Additionally, without providing information about the climate where you live, local cycling infrastructure (bike paths/lanes/etc.), or your budget, this question is nearly impossible to answer.

Possible Solutions

Nothing makes me happier than people deciding to be healthier by way of bicycle commuting, so I'd like to offer you some ideas about how to get your mother on a bike! 

Jumping into commuting can be difficult! I'm not sure how long it's been since your mother was on a bicycle, but it seems like her expectations are divorced from the reality of bike commuting. Does she ride regularly? Can you ride with her on the weekends to build up her experience with modern bicycles and her confidence on the road? I believe that she the following benefits from just getting out on the road:


Strength (no electric assist)
Balance (regular two wheels?)
Familiarity with route, weather, and mechanics of modern bicyles
Confidence


If it isn't possible to ride with her, encourage her to join a local cycling club (if available) and ride with the older group. Once she finds out whether this is something she can commit to (or not), you can start making plans to set her up with a commuting rig!

Other tips:


Buy used! (this will keep the cost of gear down)
Use a rain suit in the rain