Let him try with supervision and LOTS of reminders to be careful!
My son walked up and down stairs by 1 year (but we had lots of stairs and didn't want him to be afraid of going up and down- though Grandma and Grandpa's idea was to just gate it off and hold hands up and down even now though my niece is 4 and my son is 2 and has been doing stairs fine for over a year now)!
Jumping, my son has been jumping some since he was just over a year, but now he is starting to jump OVER things! He gathers things together on the floor and jumps over them for his baby brother to make him laugh!
My son still does not ride a tricycle, but does like the toddler powerwheels with the finger push button (though we make him still wear a helmet).
My son tried tumbling when he was about 9 months old, but my husband's parents put the kabosh on that before I could teach him to try again and do it safely. And we now live in a house with wood floors, so I don't think he wants to try, but if he wants to try, I would definitely let him and just help him discover how to turn over without hurting his neck.