I've been reading over the rules and was hoping you could clarify exactly how much of older code and assets it's okay to use in a game jam game.

I've participated in about six of these jams so far, and am starting to build up a bunch of things that I can reuse.   It doesn't really make sense for me to write a new character controller or a camera controller from scratch for a new jam when one I've already written for an old jam does exactly what I need.  There are lot of other snippets of code where if I were to write them fresh, I'd basically just be duplicating code I've already written.

There's a similar problem with assets - it doesn't make sense to make new grass or wood textures when I've already painted several.

There's also an issue with asset packs - why is it okay for people to use images, models and music created by third party artists that you can download, but also have a rule saying that the majority of assets need to be created during the jam (effectively preventing you from reusing your own assets)?

I am not mod so do not take my advice as authoritative but I do agree that it is a grey zone. Someone could have submitted a game made before if it somehow fits the theme nicely or instruct your whole game studio to work for the game. I think the intention is to encourage the creator to create as many new things as possible. If you feel like yourself simply swapping out something/re-skinning from the old game then probably it is not ideal for the jam.

My personal take is that do not see this jam as a competition and the winner should not be incentified. Try to always create a game that has some new elements in it whether it is new genre, new mechanism, new set of assets. In your case I would say just have a new gameplay mechanism is enough to justify even you reuse all the assets.

I think the thumb rule is that your game should be something new, not a game you did before. If you are doing a new game, but there are generic assets (code or not code) that you can re-use, than makes sense to use it. You will not learn more for re-implementing the same stuff. If your game is really different from your previous game and you can re-use an asset, then that assets is generic. 

For instance, I always re-use (and improve/extend) player controllers when I do a new game. I also usually re-use and adjust the main menu, just replacing fonts and background image (sometimes adding/removing entries from the menu). I've often re-used light textures also. And so on.