I have been a bespoke picture framer for the last 30 years. Working on gallery grade faming.
The mount-board (coloured cardboard) is used to highlight the picture and give it depth, also to hold the painting away from the glass and stop discoloration. If professionally done, the mount will have a slightly deeper lower edge, to balance the picture when hung on the wall. The mount-board will (should) be made from high quality acid free paper, also to stop discoloration and damage to the picture.
The brown tape at the back of the frame is essential to elevating humidity and dust entering the frame, extending the life of all parts involved the framing process.
Normally this tape is a professional framing tape, that sticks to the molding (wooden bit) but not to itself, but any brown tape can be used, self-adhesive or gum tape. If using gum-tape make sure most of the moisture has been removed before applying it to the back of the frame.
As for leaving a message in the back if the picture frame, I always loved this when reframing a picture. The history and love we can share with future art lovers, just makes the art feel so much more special.