I think Juggy’s hat is a derby that has had the brim cut up so it looks like a little spiky crown. Not something you’d see most modern kids wearing, so I’m surprised they haven’t decided to ditch it, too, to make him “with it.”
“I hope life isn’t a big joke, because I don’t get it,” Jack Handy
I already answered that question; weren’t you paying attention? 
It’s called a beanie; here, let me quote the comic book to prove it:
“Archie doesn’t know it – but I’m secretly a deputy in the Time Police! Some mystery figure in the future desinged a special beanie – so I can think my way across time!” --Jughead Jones, Jughead’s Time Police no. 1
Issue no. 6 revealed that Jughead was in fact responsible for the creation of the time travelling beanie (I’m thinking the creators of the comic book knew that it was to be the last issue so they decided to answer the one mystery they had been taunting readers with since day one).
And now in three posts I have finally made reference to every single issue of Jughead’s Time Police.
“I’m just too much for human existence – I should be animated.”
–Wayne Knight
As for Betty’s change in attitude, I remember in one of the Digest comics, a story where Betty wasn’t sure how to wear her hair to the beach, and finally decided to just wear it down. Without her ponytail, she looks absolutely stunning. Every guy on the beach wanted her – especially Archie! He was tripping over himself, cartoon hearts floating around his head, begging Betty for a date. But now Betty realized she was in the same position Veronica was in all these years, and played the same sort of mind games with Archie that her best friend was so fond of playing.
This event must have given Betty a new confidence, showing her that she didn’t need elaborate schemes to win Archie; she just needed to be herself (apparently a much less mischievous person).
no time travel was involved.
“I’m just too much for human existence – I should be animated.”
–Wayne Knight