If your operations are exclusively within one state, you can file for a state exemption number, as well as a single state fuel tax sticker.
You need to have the number displayed, and documentation of the validity of the DOT number.
should only need a dot number if it is used for a commercial purpose
Find someone who'll employ you once you have a CDL, and will offer their truck for the road test. It doesn't have to be a dump truck. You could rent a 33k single axle truck from Penske with an automatic and air brakes, and, according to the DOT, it would qualify you to drive a Class 8 dump truck.
You only need a DOT number if you own and operate a Big Truck. INCORRECT! If you're in the transportation business period, you need either a USDOT number, or, for an intrastate only operation, a state DOT exemption number. It is not limited to just large vehicles - a company such as Hertz, which rents out passenger cars, is also required to have a USDOT number.
If it's an interstate company, absolutely. An intrastate carrier can get a state issued exemption number.
There's no logo requirement on a commercial vehicle. On a Class 8 tractor--a semi--you need the license plate (base plate) and either a DOT number or "not for hire." You can put "not for hire" on a truck that is not used in commerce, like the tractor that pulls a race team's trailer. Everyone else must have a DOT number. But as far as logos go, you don't need to have a logo on your truck.
Dot 3
DOT stands for Department of Transportation.
It is a DOT hazardous materials placard indicating compressed oxygen.
You'll typically need your fuel tax sticker (either IFTA or state), your company's DOT registration number or state exemption number, and some states may require that you include the vehicles gross weight rating on there, as well. Ultimately, on a matter such as this, you need to contact your state's DMV, and regulations on intrastate carriers can vary between states, and there may be additional requirements in your state which I missed in this answer.
Laws on this vary by jurisdiction. You need to contact the DOT/DMV of your state to determine this.
You can't. The DOT number pertains solely to the trucking company, as does the ICC/MC number. If you have their truck number, and have a complaint to file against that driver, you can call the company and they'll know who to go to, but they're not going to give out that driver's personal information.