How many nights in Amman?

One on the way in, maybe one on the way out. More only if the north is a real focus or your flight times force it.

Can I climb in the city?

Indoor walls exist. Outdoor limestone and sandstone are south and in the rift canyons, not in downtown Amman.

Is it a good base for Mujib?

Possible as a long day. Kinder to stay closer, in Madaba or at the Dead Sea, if the Siq is the point.

When should I avoid a north day from Amman?

Midsummer midday, and Fridays if your operator warns about traffic and site hours. Early starts fix most of it.

Things to do in Amman

Amman is arrivals, traffic, and the night before everything else. City walking and a few indoor walls exist. The outdoor trip starts when you leave. Use it as a launch for the Dead Sea, Jerash, or the south, not as a week of capital sightseeing dressed up as adventure.

First night and day trips

Land, sleep, then go. A Dead Sea afternoon or a Jerash day is a fair use of a spare 24 hours. Don't try to reach Wadi Rum for sunset after a midday inbound. Open the Book here tab for trips that genuinely meet in Amman and the north.