For inflexibility:
* Flexibly
* Adaptably
* Elastically
* Yieldingly
* Pliable
For strictness:
* Loosely
* Casually
* Relaxedly
* Openly
* Permissively
For formality:
* Informally
* Casually
* Relaxedly
* Spontaneously
* Improvisational
Choose the best opposite based on the specific meaning of "rigidly" in your context.