I just finished learning 100 egg, and I'm practicing before doing runs. But when I was looking at the rules I saw "Easy-Mode throughout the whole run is not allowed".

I know dying in Sunny Villa can activate easy mode, but is there any way to disable it? Does this mean dying in Sunny Villa can invalidate your entire run?

So basically, if I'm not using a memory card there's nothing to worry about?

yes

Pretty much. Whenever you do runs of 100 egg and 117%, the first level you enter will ALWAYS be on easy-mode. After you've collected a certain amount of eggs and gems (it's barely anything you need to collect) it will activate hardmode. So there's nothing to worry about if the first level you enter is in easy-mode.

~KJ