Should I have a pair of shoes that I wear only in the lab?
Standard safety rules (from a pharma and semiconductor processing POV):

To follow industrial hygiene practice absolutely to the letter, you should have dedicated outer layer garments that remain in a quarantined area that would not pose a gross cross contamination risk to other things that cannot be controlled for exposure risk. This means that you should not have been able to take those boots home with you after you wore them in the lab even one time. I do not think any company enforces such a rule, in fact, it is the employer's responsibility to provide PPE to workers and the worker's responsibility to use and maintain PPE (if applicable.) If the risk of cross contamination was assessed and determined to be moderate or higher, you should have been provided shoe covers at your solid state systems lab or dedicated work shoes that had to be kept in a quarantined area.

At the lab bench level, it is my experience that people wear their daily shoes unless you work at a larger site that has a safety shoe/shoe covering policy.