Sadly, those characters only give you one chance to recruit them. If you don't have the correct person talk to them, then you can't recruit them.
Note that this is true even in Casual mode; Casual only flags your characters as unable to die. Until you recruit them, you can definitely kill them, so be careful if you're trying to get everybody.
One thing to do at the beginning of each mission is to run through every enemy; a recruitable characters will have a name and a unique portrait, and not be the boss. Bosses cannot be recruited except through story means, so there's no need to be careful with them.
When you find a character you can recruit, tap their name; it will often give you a hint as to who needs to talk to them to recruit them. Usually it's Chrom, but it might be someone else.
Awakening also has a neat feature where if the person who can talk to them is within movement range, they'll have a "Talk!" icon above their head when you try to move that person. That's new to Fire Emblem; previous versions meant you had to move a person beside the recruitee, and see if a Talk option was available. If not, cancel, and choose the next character. That still works in Awakening, but they make it easy on you with the icon.
No, you will only miss out on one thing by playing on casual mode. If you beat the game while playing casual on the Lunatic difficulty, the NG+ for Lunatic will be casual only. This is different from beating it on classic mode which unlocks both. This is also different from the other two difficulties where beating it in casual unlocks both classic and casual NG+.
Other than that the games are identical except in casual mode permanents deaths are turned off and you can save in battle, you're allotted two slots for this. There are no extra boards/characters/etc. that can only be obtained by playing on classic.
Best Answer
Yes, the story of Awakening (FE 13) is relatively stand alone, meaning you'll understand the story without having played any of the previous installments.
However, Awakening has a lot of references to earlier games. The most prominent are:
Chrom, one of the main characters of Awakening, is the decendant of the Hero King Marth1.
My Unit (default name: Robin), aka. yourself, and another main character of Awakening. Not exactly a reference to anything, but...
Anna, who used to be a recurrent NPC in most Fire Emblem games and is now a playable character.
Tiki1, who has grown into a fine lady since her adventure with Marth. Well, she is a divine dragon with a much longer lifespan than any human.
Priam, who claims to be the decendant of the Radiant Hero, Ike2. He also carries Ike's sword, Ragnell.
There are also many smaller references, such as Alm's Blade3, the Deadlords4, and many more.
Knowing all these references is nice to have, but will not give you any advantage in understanding the story or playing the game. It is therefore safe to play Awakening without ever having played Fire Emblem before.
1: FE 1 & 3, as well as their remakes: FE 11 & 12
2: FE 9 & 10
3: FE 2
4: FE 4 & 5