You are always free to do what you want to your product and if you decide you want to advertise your company name or website on there you can do so.
There are a number of different methods to add graphics to your product, typically these are done using pad printing, screen printing, stickers or simply add the graphics to the tool so where it can be removed if necessary.
If using colour graphics try and keep the number of colour to one or 2. The more colours used in the above methods will add more cost. Adding the graphics to the tool will cost you during the tooling phase but after that there is no additional cost. Be aware that if you machine this graphics in the tool it will have a lot of detail and will potentially wear quicker than any other part in the tool so replacement graphic inserts may be required.