Yes — if the design was sublimated. No, reliably, if it was a decal. The difference is invisible in a photograph and obvious after a month.
A sublimated mug is dishwasher safe and microwave safe. Under heat the ink turns to gas and bonds inside the glaze, so there is no printed layer sitting on the surface that could lift, and nothing metallic to arc in a microwave.
A decal or vinyl mug is a different product that looks identical online. The design is stuck onto the finished glaze, and dishwasher detergent plus 60°C water works at that edge until it lifts. This is why so many novelty mugs end up hand-wash-only after their first month.

Very little, and none of it is the dishwasher. The realistic threats are abrasion and thermal shock. A scouring pad dulls the glaze over time, and the glaze is what carries the image — once it goes matte, the print looks faded even though the ink has not moved. Pouring boiling water into a mug straight from a freezer can crack ceramic, which is true of any ceramic, printed or plain.
No. Microwaves heat water, not pigment, and the pigment is sealed under glaze. What people remember as microwave fading is almost always a decal mug lifting at the edges from repeated heating and cooling cycles.
The test that settles it: put the mug through fifty dishwasher cycles. A sublimated print looks the same. A decal tells you long before fifty.
Longer than the mug, in normal use. Sublimation is the same process used for commercial signage that lives outdoors, and inside a kitchen cupboard it is barely stressed. Mugs in this shop are printed individually after the order arrives, then packed in moulded protection — if one arrives broken, a photo is all we need to remake it.