Take a look at possibly connecting your Blu ray player with the Analog connection. If your neighbors had analog, then the DVD has Analog Subtitles on it instead of Digital.
Connect that video cable connection to your Side panel --AV2
Now, while we are at the TV set --
Now those setting most likely ONLY work with the RF input on your TV set, and you may want to try setting SUBTITLES on your Built-in DVD player WHEN YOU HAVE A DISC INSERTED INTO IT !
Try to access the DVD's menu BEFORE and AFTER the DVD is inserted.
Now there may be some subtitle setups BEFORE ....and possibly AFTER inserting the disc you want to watch with subtitling.
When you set your settings BEFORE -- that actually sets up the DVD player, and when you Set it AFTER inserting the Disc, it Sets up the DVD disc !
That may be why your TV couldn't get the subtitles off your DVD.
As for the Blu Ray....we'll deal with that.
Set your Blu Ray to use BOTH the HDMI and the Composite Video Cable. I suspect the HDMI cable can only deliver DIGITAL CC and you need ANALOG CC off of the Disc for it to appear.
Let's deal with the TV setup first. Let me know if you are successful with those settings.