Do all cells have the same DNA?
All cells in a person’s body have roughly the same DNA. As you point out correctly, mutations can happen in any cell at any time. They are frequently caused by


UV radiation
Chemicals (think of the correlation between the development of lung cancer and smoking)
Replication of DNA during cell division
Integration of foreign (e. g. viral) genomic material
On purpose by the body’s own enzymes


That last point brings us to cells of the immune system, some of which (classically T- and B- Lymphocytes, but there is emerging evidence for others) generate variance within their genomic DNA on purpose.

Red blood cells lack a nucleus altogether.

Gametes only have half the body’s genome, as you already pointed out.

Finally, although having the same base sequence, epigenetic modification of DNA molecules by, for example, Methylation makes DNA chemically different even between cells of the same type.