Do tickets expire

It's not in the Mii Maker, which is kind of counterintuitive:

The special Mii hats can only be used in StreetPass Mii Plaza. The "Mii Settings" that you're looking for are not present in Mii Maker. Select your Mii in the StreetPass Mii Plaza and you'll see "Mii Settings", where there is an option to wear a hat that you found in Find Mii.

You can safely ignore the gathered StreetPass data for weeks at a time. Keep in mind, however, that while StreetPass Miis will wait patiently at your gate until you check it, that only 10 StreetPass tags can wait in line at a time. Also, if you encounter the same 3DS unit multiple times - Mii Plaza will acknowledge that you've tagged them more than once (and this does not seem to count against the 10), but you still only get one use out of them in each game.

You can check how many Streetpass tags are available for being encountered at your gate through "Notifications" on your home screen (between the friends list and web browser). This should also clear the green notification LED. Once Notifications says you have 10 Streetpass tags for Mii Plaza, you should check them before any others are turned away.