Stickers are a way of taking things you're already probably familiar with from other TCGs (eg. special event cards, member cards etc.) with the one difference that they are not cards and are totally optional. If you don't like member cards and other extra site related cards you can totally ignore all stickers without it negatively effecting your card count or participation. If you do like those kinds of things, then stickers are a fun way of tracking your membership at 5"x7.5".
No. Stickers track your participation at 5"x7.5". If you don't enter a contest, you can never own a sticker for entering that contest. If you miss a certain sticker, there is no way for you to go back and get it later.
You can trade member stickers with other members.
Nothing. Stickers are not cards. You can't trade them for cards and trading member stickers with other members doesn't count towards your trade count.
Joining earns you a member sticker, plus a sticker showing your member status (either member or prejoin) and one with your member number.
Your birthday every year gives you the opportunity to apply for a special birthday sticker.
Referring members earns you a sticker every time you refer a certain number. If you manage to refer 50 members, you'll also gain a patron sticker.
Trading with members earns you a sticker every time you reach certain milestones. Each card is counted as one towards your trade count. Trading 1000 cards earns you a trader sticker.
Donating to 5"x7.5" earns you a different set of stickers every month
Participating in & winning contests gains you a sticker that tells you where you came in each contest.
Participating in & winning Cardmaker rounds gains you a sticker that tells you where your entry placed.
Special events are often marked by stickers - for example, site anniversaries and holidays!