The good news here is: AIX is a UNIX that is very nice to administrate even for beginners because of the System Managment Interface Tool SMIT. Nonetheless you might hint to your boss that you cannot take responsibility for a production server without proper education.

The mirroring is visible by the number of Logical Partitions LP in relation to the number of Physical Partitions PP. There are twice as much PP as LP and thus you know the Logical Volume LV consists of two mirror copies. This is transparent for the application which sees just the LP.
As to the second allocation problem. Whenever you get problems with the allocation it is a good idea to look into the LVs configuration.
# lslv <yourlv>
and into what the Volume Group's disks look like. Allocation problems usually result from the LV configuration requireing a distribution of the data that cannot be fullfilled with the diskspace available. You can then either change the LV's configuration or make the diskspace fit (by adding disks or migrating data). Most of this work can be done online.

A mirror makes sense only when the PPs are placed on different disks. While you could place two copies on one disk the default is to force using different disks. This is likely to be the case with your /usr filesystem. However from the lsvg output it is not clear whether there is the same number of free PP on any rootvg disk. So after checking the LV you need to find out where you could possibly place the data. There are several commands to accomplish this but for now you might use
# lspv <rootvgdisk1>
# lspv <rootvgdisk2>
to find out how many PP are free on your rootvg disks. If every rootvg LV is mirrored evenly there should be 11 PP free on every disk because there are 22 PP free altogether in the VG. However, as there is another alloc Problem the PP might be distributed unevenly among the disks. So the trick here is to extend the /usr filesystem just by the minimum number of PP that are free on every rootvg disk. E.g. should there be 6 PP free on hdisk0 and 16 PP on hdisk1 you extend your /usr FS by 6*32M.

In case there are no free PP on one disk at all or not enough PP it is possible with AIX to move PP from one disk to the other online. In that case come back and post the output of lsvg -l rootvg.

HTH