don't know that there's a single mat made to do all you are requesting, but here are a couple ideas.
regular mouse/desk mat
https://www.amazon.com/Extended-Stitched-Mousepad-Premium-Textured-Waterproof/dp/B0788LMLZLthese are a popular item on just about any keyboard vendor's site tons of designs, usually to go along with a particular keycap set.
- fabric & rubber backing - can't solder over it, not that nice to write on, con't cut on it (will cut the mat as well) nice for mouse & keyboard to sit on.
dedicated soldering mat.
https://www.amazon.com/Kaisi-Insulation-Silicone-Position-Soldering/dp/B07XTDGN4T- usually a blue silicone mat of various sizes - made to solder on usually some sections on the mat for small electronic parts, can't say how a mouse will be on it, probably not that nice to write on, probably easy to cut. (i don't have one)
cutting mat.
https://www.amazon.com/OLFA-Double-Sided-Self-Healing-Rotary-Mat/dp/B0006SDOFO- made of some types of plastic/vinyl made for cutting on, ok for a mouse, will leave heat / solder marks where solder hits it, but solder peels up easily, this is a harder mat, fine to write on like any hard surface.
- this is what i use for my workstation - cutting/soldering/tinkering etc... i have a large one 24"x36"
for what you're describing... i'd go with a cutting mat.
hope this helps.