You cannot merge them in any automated fashion. You have to manually copy one into the other.
btrfs provides some tools here though; you could use btrfs send | btrfs receive
to transfer the subvolumes for instance.
You'd transfer all of your subvolumes in the root partition over to the "home" partition and then adjust your fstab accordingly. After a rebuild of the initrd, your system should boot into the root subvol of the formerly "home" partition. Once that works, you can delete the old root partition and extend the new combined partition.
You can generally do almost anything to a btrfs while the system is running with the filesystem mounted. In fact, most operations require the btrfs to be mounted.
You should keep a live USB handy though in case you mess up. Oh and also verify that you have a sufficient amount of backups of your important data. 1 copy can turn into 0 copies quite quickly. Rule of thumb places the minimum at 3 copies.
Yes, for modifications to the root mount, you must rebuild the initrd for those changes to actually be applied where it matters.
How this is done fully depends on the distribution and is one of the more significant technical details that differ between just about every distro family. Consult your distro's documentation on that.