Hi, is it possible to increase the space of the local-lvm which is on a partition using the space of a newly added disk? In a nutshell, looking at the picture, can I merge /dev/sda and /dev/sdb3 under one volume without losing data?
You can. You would need to expand the Volume Group with a new Physical Volume (sda), then expand the LV.
It may not be wise to mix too dissimilar disks though, besides strange performance disparities a disk failure may lead to corruption of entire dataset. Especially considering that new disk is not protected in any way.
plug this to google "extend volume group with new disk"