LVM provides logical volumes, which are logical block devices, and makes it easy to grow, shrink, relocate, snapshot, etc. those block devices. You can then use these block devices any way you like... it could be a filesystem, or something else like a virtual HDD for a VM with its own partition table and everything.
LVM does nothing on the filesystem level. So it's up to the filesystem to support handling those grown or shrunk block devices, or to the VM to resize their partition table.
Most filesystems support growing (but sometimes not online, or not past a certain limit), but a few of them don't support shrinking. So although LVM has no qualms about shrinking the block device, you'd have to shrink the filesystem first and for some filesystems, that just isn't possible.
Merging contents of two separate filesystems is usually not supported.
So yes, in some cases, you have to copy files the old-fashioned way. And then abandon/remove the LV those files were on, and use the freed space to expand the LV and grow the filesystem you copied the files to.
So how can these be combined? Is that advisable? (because of different uses for SSD and HDD)
I would not create a block device that is backed half by SSD and half by HDD. I like to keep these separate.
It might make sense in some other situations, e.g. you can do a SSD-HDD-RAID1 where the HDD is set to write-mostly, which means all reads will normally be served by SSD as it's faster. However with dropping SSD prices, that setup is less common as you can just use two SSD for regular RAID1 instead.