removing a plugin

I am not sure if this is the correct forum to place a general plugin removal troubleshooting question. If it is not, please let me know and I’ll close/move it. I have attempted to contact the plugin author without success. At this point, being a relatively new to all this, I just need some advise on how to proceed.

Prior to installing the plugin, all aspects of my website were functional. I install the Custom Contact Forms plugin and it caused a conflict with other plugins that used reCAPTCHA. I decided to deactivate and remove the plugin. At that point, my Google Adsense ads were no longer visible on any pages. Through a lot of troubleshooting, I found that if I re-installed and activated the plugin, then deactivated but not delete it the ads reappeared. I am sure that it is a Java Script issue.

1) Is it a problem to leave the plugin installed, but not active?
2) If I want to remove the plugin, how might I troubleshoot to find the root cause of the issue to fix it.