Various theme coding issues.

I’m sorry if these codes are addressed somewhere but I’ve been reading through everything since 3 and it’s 6 now and although I’ve solved most of my own issues I just can’t find the right combinations for the last few.
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First:
I’m trying to alter my category template to adhere to the following logic.

if this is a parent category list the children of self
if this is a child category list the posts within self
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Second:
Is there a code that, when viewing a single.php generated single post, will echo the category title or category description of the category the post is listed under?
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Third:
Is there a code that, when viewing a page.php generated page, will echo the page title? (I know this one should’ve been easy but I just can find it.)

Thanks