Dental implants will be done for 2 side-by-side missing teeth. Can the adjacent teeth shift out of place since cannot wear removable denture anymore?

Unlikely. After the implants have placed and integrated with the surrounding bone, the adjacent teeth shift cannot shift. You can probably have an orthodontic retainer made to wear; this prevent the opposing teeth from shifting.
See your dentist. if the tooth is mobile or has premature contact, it may cause movement of the tooth. Have your dentist evaluate your bite and see if any adjustment or refitting of the partial denture is needed.
Maybe. How long before the implant surgery? If you will be waiting for over 6 months until implants are placed, and they are not restored right away, yes, the teeth can shift. I recommend getting a temporary removable appliance like an Essex retainer or an interim denture to hold teeth in place while the implants get integrated. Talk to your restorative dentist if it is not the same as the implant one.
Possibly. If it will be problematic, wear an orthodontic retainer at night. The Vivera (Invisalign) retainers will not impinge on the implant site.