How do you download your videos? if you download your videos as .flv or .mp4 files using any kind of video file downloader, then no, you're safe, because once your download is finished and stored in your hard drive which you picked to download it into(assuming that you are talking about manually downloading the video file), the file is stored into your pc and you are to decide if you want to delete it or not. youtube not supporting your browser simply means that once youtube has updated it's structure, the website will not load properly on your browser, as in you won't be able access some of it's new features, the website's layout may be off/weird, videos may not be playable in your browser and buttons may not work etc., nothing serious. this doesn't mean youtube will do anything to the files you already have downloaded, as it does not own the right to do so.
If you have an account on youtube and favorite/like videos, you can use any other browser, log into your account from that browser and find all your favorite videos as this information is not stored into your browser but into your youtube account. making playlists and storing videos inside them is a good way to keep a list of videos you'd watch often, that way, no matter where you are, if you simply go to your youtube channel, you can watch any of your favorite videos at any given time.
And if files DO disappear on your pc without your permission(given that you have saved those certain files yourself, manually,) it may mean that someone else or a virus(most possibly not) is deleting them...that's the only conclusions i can come to.
When you load a youtube video while using your browser and watch it online, it is stored as a temporary file in your C:/ drive, that file can indeed get deleted and often does get deleted once you exit your internet browser or refresh or even just leave youtube to keep good pc speed and space, but that does not mean that the video will be gone from the website itself, it will remain there as long as the uploader did not decide to delete it.
Good luck with your files and your browser.