Clash Royale - Can you get unobtained troops from previous arenas?

Yes it is possible to get troops from previous arenas, you can get cards from the arena you are currently in as well as all the arenas below you. If you are in arena 11 for example, you are able to get EVERY card in the game because arena 11 is the highest arena. An arena 11 chest you earn from battle has a chance to drop any card in the game, but an arena 9 chest can only have cards from arena 9 or lower. The only exception to this rule are legendary chests, which can drop legendary cards from ANY arena.

The first way to get cards is to open chests(Daily,Crown,Silver,Golden, etc...)

The second way is the play tournaments, which sometimes you can find free tournaments to join under the tournaments tab, they are usually only 50 player tournaments but you can find a free one if you fiddle around.

The third way is to buy cards from the shop, starting in arena 10 Hog mountain you can even buy legendary cards which will be purchasable for 40,000 gold

The fourth way(MOST EFFICIENT FOR YOUR TIME AND MONEY) is to play Challenges, meaning GRAND CHALLENGES, if you truly have the skills and want to get cards, 100 gems to enter a grand challenge can get you a lot of value. If you average 6-8 wins then you can get cards/gold easily. Basically you play these until you get 12 wins or 3 losses against random players, exactly like Arenas from Hearthstone if you know what they are, but much more skill based than HS.

Source: Clash fanatic, been playing Clash Royale since release