Is it permissible for someone who arrives early at the mosque to reserve a spot in the first row and then engage in other activities outside of the row or the mosque?

The first issue: Reserving places in the mosque with prayer rugs (sajjada) or the like is a matter of scholarly disagreement. The Shafi'is and Hanbalis permit it, while the Malikis and Ibn Taymiyyah prohibit it. The more correct view is its prohibition, due to the inconvenience it causes people and the need for others to step over their necks.

The second issue: If someone leaves their place in the mosque for a need and returns to it before the prayer is established, they have a greater right to their place than others, even if they did not leave anything there. They also have the right to ask the person who sat in their place to move. This is the correct view according to the Shafi'is and is the madhhab (definition) of the majority of scholars.