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I think normal lotteries and chain lotteries deserve separate sub-categories. They are really very different things, and I feel the proliferation of chains has made the lottery section cumbersome, and possibly confusing/intimidating to newbies.
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Perhaps an “other postal games” category would also be in order. The guessing games that are currently in with lotteries could be moved there, and it could also be a place for any new game someone might invent, that doesn’t fit neatly into another category.
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People seem to be using the “Random acts of Smileness (RAS)” sub-category the same as the “Spread the Joy (non-RAS)” sub-category that can be found under Trades Requests and Offers, and it’s confusing to have those type of requests spread over two different categories. My suggestion would be to remove the RAS sub-category from Games and Activities (of course moving any active topics in it to “Spread the Joy”) and just pin the original RAS Round Robin (which shouldn’t need a whole category to itself) wherever it fits best. I would think under “Spread the Joy” (with the “non-RAS” part of that title removed, of course!) but if people are attached to keeping the RAS RR under “Games and Activities,” it could also go in the “other postal games” sub-category I suggested creating above.
That’s a thought I had as well. I used to like to take part in lotteries, but now the entire section is full of chain lotteries, which I don’t like at all. It would be quite helpful to have them in different categories, especially as not every one of them has “chain lottery” in the title. I have almost given up looking for lotteries I might enjoy.
I can’t say anything regarding your other suggestions, as I never visit this part of the forum. 
Regarding suggestion no. 3 I think it should stay at it is.
to 1: For me personally, it is not necessary to separate the different types of lotteries.
I don’t generally take part in many lotteries, but only if I particularly like the winning card.
If it is a normal lottery - good.
If it’s a chain lottery, then I just open a new lottery (if I’m the winner)
But then I turn it into a normal lottery, because I don’t like the idea behind chain lotteries.
For me, the point of a lottery is that I want to make other people happy, and for the winner that means that they can simply be happy without having to do anything for it.
to 3:
As @Nordbaer said: Leave it as it is.
Is that allowed? I thought one of the requirements was that the new lottery the winner opens must also be a chain.
I never participate in chains, because when I host a lottery I like it to be open to everyone, whether or not they can host one of their own. If I knew it was okay to end the chain with a normal lottery, I would go ahead and participate. Maybe if more people did that we could thin out the chains some 
I still think separating them would be better though, personally.
And as for point 3, is their something I’m missing? Is there a reason to spread those requests across two different categories, that I’m just not understanding?
I used to only check for them in “Spread the Joy” every time I came to the forum, and sometimes I would even see something in suggested topics and want to send a card to the person, but look for the post later in “Spread the Joy” and not find it 
Then I finally discovered how many requests for cards to cheer people up were hiding in “Random acts of Smileness.” So I’m just trying to understand why it needs to be that way.
There is no official rule on this.
And the rules in the 2 or 3 Chains I won only ever said that I had to open a new lottery if I win, which I did.
I agree to this. It will keep things neat and organised.
Another option would be to create defining tags for lotteries, like “normal” and “chain”. That way people could be able to filter with their preference. There is already a tag “guessing-game” in the lottery sub-category. Personally I find it good that all lotteries are in the same sub-category, but using a tag would help people to find the “normal” lotteries more easily.