(1) Expect the initial Part 385 "Safety Audit" to occur in three to six months after initial DOT# registration. But, in event of involvement with a bad accident, it could happen tomorrow.
(2) Everyone with a commercial vehicle (per Part 390) needs to comply with 24/7/365 Hours of Service rules: logbooks are a must over 100 air-miles, and 7 day/week "exemption sheets," are needed under 100 air-miles (if all other qualifications are met). Looks like you will probably need both if you occasionally go beyond 100 air-miles.
(3) While the DVIR is not "technically required" for one vehicle, the DVIR protects both the company and driver by documenting all inspections and repairs. ("If it isn't documented, then it isn't or wasn't done.")
Logs, DVIRs and otherforms are available at most truckstops or from
www.jjkellercom; call their toll-free number and ask for their "Transportation Catalog."
All the best