Learn English – “Give someone trouble” — what does it mean

This phrase is a modification of the crude, informal present-day phrase, "You can stick it up your ass."

This phrase indicates two things: The person speaking does not care about it, and the person speaking does not like the person who is being targeted in the phrase. "Carry it with oneself always" is a good guess, but this phrase doesn't have that meaning at all.

The use of "They can stick my application form up their jumpers" is used to say that the not only does the speaker not like the people who have his application, and he also doesn't care about the application. "They can stick my application form up their asses" would have been about the same thing, except that in the phrasing the speaker used, he's probably making fun of them for wearing jumpers, too.

This phrase is very rude and not a very good "joking" insult -- it should only be used when you really don't like someone!

The humor comes from the fact that the speaker has used this phrase to declare his contempt for the people involved in a round-about way. I wouldn't say it deserves bursting into laughter, but there you go.

A sniff (drawing air through the nose in a short, audible inhalation) is sometimes used as a sign of contempt. See Dictionary.com:

4. to show disdain, contempt, etc., by or as by sniffing.

Nynaeve is disparaging Elayne's comment by the sniff. Note that a sniff can be a haughty way to express disdain, as if words would be too much to waste.