The robot can carry small weights on its own, yes, although it's way more stable when the arm is retracted closer to the body.
Pick-and-place style problems are one of the earliest tried on our robot platform, Hello Stretch, which is why this project doesn't spend too much time on it. You may enjoy something like the OVMM project [0] which focuses entirely on open-vocabulary pick-and-place problems: "Pick up $A from $B and put it in $C".
