There are no "old" updates and operating systems - when the operating system gets updated, the old version of that part of it is overwritten by the new version.
As usual, that's another "Verizon tech", really a sales associate, who doesn't know what he or she is talking about.
You can't "remove the old stuff", the update did that already. The phone has a total of 16GB of storage, which is pretty small these days (because apps keep getting larger). You could, if you wanted, back up all your old stuff, reflash with an older version of the operating system, root it so it wouldn't update (even if you tell it to not update, after about 15 times it will anyway), then restore all your stuff.
Or you could just add an SD card to the phone, and put all your music, pictures, videos, etc., on the card, and delete them from the phone. (Don't use Move - if that blows in the middle, you could lose both the original and the copy. Use Copy, then Delete the original once you've determined that the copy is good.) You'd still have them, but you could have over 200GB of storage space for them - now [the phone can take up to 2TB cards, they just don't make them that large yet]. (Not for apps, though - you're still limited to about 8GB [the operating system, which is Linux, and Android, take about the remaining 8GB] for apps on that phone. Moving apps to the SD card is a bad idea for technical reasons, and it doesn't save you much space anyway.)