Rule book: http://www.hasbro.com/common/instruct/Risk_2003.pdf
The 5 Territories Mission
There are three possible interpretations:
The player needs to have conquered all of the territories this turn. This has the problem that a player who controls two or more continents cannot fulfill this mission. I believe the rules do not support this interpretation.
The player can have conquered the territories at any point in the past and that's it. This has the problem that a player who has fulfilled this condition at any point can claim this mission at any later point. This has tenuous rules support.
The player can have conquered the territories at any point, but must still control them to claim the mission. I believe this is the correct reading, but my argument for this over option 2 is weak.
This mission does not specify that the conquering all needs to be done in a turn. It simply means you have to have conquered a territory on each of five continents at some point already. It does seems like a bit much bookkeeping to me, but that's how the rules and cards read.
Here are the relevant rules:
Page 12: You win the invasion when you defeat the last defending unit from a territory. You have now conquered this territory. Take the units used to win the battle and move them from the battleground into the territory you have invaded.
Page 17: You may claim a mission once you have achieved what it says on the card. But you cannot claim more than one mission a turn.
At the very minimum, we can conclude that you can claim a mission even if didn't take all of the actions required to complete it this turn. This must be true, or else the mission "Control Europe - play at the start of your turn" would be impossible (as this is a mission where you need to have taken the actions to bring you to the state where you controlled Europe on a previous turn).
Given this mission does not specify whether the conquering needs to have happened this turn, then the conquering certainly could have happened in the past. This rules out interpretation 1.
The question then is whether your status of having conquered a territory is an indelible event in history (interpretation 2), or if it's tied to the current game state (interpretation 3).
I prefer interpretation 3, so I'm going to try to create a rules justification for it, but I admit that this is a bit of a stretch. The justification for this is a nuanced reading of the term conquer. The rules for conquer require you to move units into the territory, so I argue that if you no longer have units in the territory, you lose your status as its conqueror. If this is true, this mission says "conquer" and not "control" to prevent players from being credited with territories they claimed during initial setup (which are "claimed" and not "conquered").
The Continents Mission
Page 8: If you control every territory within a continent, you control that continent.
Page 7: territories you control [are] the territories with your units in them
Note that this is a deviation from classic risk which has the following:
To control a continent, you must occupy all of its territories at the start of your turn
In the rules supplied with mission risk, you may conquer the continents that turn and still have that count towards the mission. In fact, it should be obvious from the fact that you claim that mission at the end of your turn that you can have conquered the third continent on the same turn.