Are there any tools / "life hacks" to cut sphere-shaped potatoes and carrots?
I think @ChrisH is on to something with the melon baller. They come in various sizes such as 1 cm, 1/2", 3 cm, and 1", to name a few. While 1 cm is the smallest I could find, it will still be larger than the peas which have an average size of .5 cm.

That said, you could use the 1 cm or 1/2" melon baller for the rest of your veggies and the egg. Of course, the egg will have to be cooked before scooping. But I think that is the answer for the carrots, too. Clean them and cook them whole before scooping.

While you could go either way with the potatoes, I think I would peel and cook them first.

With the harder veggies, you want to be sure they are fully cooked, but be sure not to overcook them. Too soft and the scooping could be a disaster. 

The other option for the harder vegetables would be to partially cook them, scoop the balls out, and then finish cooking to your desired doneness.