Conversion is a manual process.
/boot cannot be on an LVM volume and there is little or no benefit in putting swap on an LVM volume. Unless / is large and has changing data you need to back up regularly, there is little benefit in putting it on an LVM volume and some cost in administrative inconvenience.
For the rest there are reasons for and against. The "for"s are more relevant on a high-availability system. The "against"s are administrative complexity.
On a system where down-time is acceptable the major benefit is being able to take snapshots for backing up. If you have data that needs to be consistent across several files and you want to update that data during backup then that's essential.
If you want to be able to re-size file systems while they are mounted then, to take advantage of LVM's ability to change partition sizes on the fly, you need to use a file system that can be re-sized on the fly. Not ext3.
It looks like you have several file systems already, one of which is 112 GB. That's not going to fit in the 15 GB you have configured for LVM use! So you would need some free space to give to LVM before you could convert all your file systems -- or copy off to backup, take the space for LVM and copy back.