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Basically, you will get all the rewards pertaining to the rank you bought. If you bought GOLD (and above) you will get the gold rewards as long as you play the new required amount of games.

Furthermore, you will only get the rank border you see on your loading screen for the game mode you have that rank.

For example, if you finished with Gold on Season 3 in 3v3 ranked teams. You will see the gold border only on 3v3 ranked teams and not in 5v5 rank solo queue games.

From their Season 3 Rewards FAQ:

"The loading screen border trim you earn will display when playing in the ranked game type you earned it in. For other queues: In 5s normal or bot games the highest of your ranked 5s or solo / duo queue will be displayed. In 3s normal and bot games, borders earned from your ranked 3s will be displayed. For ARAM , custom, and dominion games, the highest of any ranked queue will be displayed."