Will an internal optical drive from a non-Apple manufacturer be fully functional in a MBP 13" Mid 2009?
A SATA drive is a SATA drive.  The fact that you were able to put in a SATA optical bay caddy for an SSD is evidence that an aftermarket SATA drive works.

Looking at the SATA connector of the "SuperDrive" you will notice that it's the exact same SATA data and power connector of pretty much any other SATA drive



So, the trick here is finding a SATA optical drive that has the SATA connector in the same location.

There is no reason for a different CD/DVD drive won't work for recovery.  As long as it can read/write, you can use it.  You can boot from CD/DVD by holding down C while booting.  After all, the MacBook Air requires you to use an external (USB) drive if you want to do recovery.

Speaking of recovery, if weight is an issue, why not get a USB flash drive.  You can get a 128GB USB Flash for around $30 and is great for backup/recovery. 

Personally speaking I haven't used a genuine Apple Super Drive in years - I either use  external hard drives or USB flash drives.  On the rare occasion I need a CD/DVD drive, I have a Plextor USB CD/DVD RW drive which works perfectly.