Failed First DOT Inspection. Now what?

Well see, you were working. Were you driving that rental because you wanted to drive or to return it after being used? Anything related to the business is On Duty. As for mounting the fire extinguisher, well the regs specifically state it has to be mounted. Just sitting loose, even under a seat, isn't being mounted. That one sucks he wrote you up for, but it is valid. Enterprise is simply wrong.
Here is the wording from 393.95. (Red by me)
Condition, location, and mounting. The fire extinguisher(s) must be filled and located so that it is readily accessible for use. The extinguisher(s) must be securely mounted to prevent sliding, rolling, or vertical movement relative to the motor vehicle.

Not trying to sound condescending or anything, but you really need to go get the green book and read it. Specifically parts 393 and 395.
Here is an on-line version.
https://www.ecfr.gov/current/title-49/subtitle-B/chapter-III/subchapter-B

 

I'll let someone else that has more knowledge answer the insurance question.

 

Most likely will raise your rates but I am just guessing as I have not had first hand experience. If you was not working why are you driving a truck with DOT numbers on it? Just asking DOT to mess with you in m mind.

 

Welcome to the wonderful world of trucking.
The law is right, you’re wrong.

 

I would just correct infractions and go on with my life. Nothing much you can do about it now. It is not a traffic infraction that you can defend in court, for all I know. I mean worry about the renewal when the time comes. You may want to consult with you agent much earlier than the renewal to feel them out but the underwriters usually don't offer renewals until only a couple of weeks before the policy expires, so I would be proactive earlier. Don't be discouraged and don't panic yet, things like that may happen to all of us every now and then. You may think you have everything in order until a jackas$ officer will spoil your day.

 

I have heard of guys getting tickets even if the fire extinguisher is mounted but the clamp not secured tight, or unbuckled

 

This happened to me. The guy checked over the entire 18 wheeler and my logs with no issues. This was the only thing he could find so wrote me a warning for it being mounted, and even with the clasp over it, just not latched. Ridiculous. But my fault. And crap happens.

FWIW my insurance went down about 20% a couple months later at renewal. Out of 4 inspections that has been my only documented warning / issue.