Which particles can be omitted from sentences?
は, が, and を are often dropped; に sometimes. か, as a sentence-final question particle, can be replaced with intonation.
Does the omission of particles make a sentence informal/impolite?
Informal yes, but not necessarily impolite. Dropping particles is only for spoken Japanese, so you won't see it in (proper) writing.
When can particles be dropped? (e.g. How can you decide that it's okay)
When the situation allows. Speaking informally to someone above you or with whom you don't have a close relationship is impolite. But if the situation lets you speak informally, you can.
In situations where polite Japanese is called for (speaking to your boss/teacher/doctor/etc or giving a formal speech/presentation), all particles must be kept in the sentence. Omitting particles does not change the meaning of the sentence or make it incorrect per se, since the missing particles can be inferred from context and word order, but it does make the sentence informal and thus unsuitable for polite contexts.
As noted above, in writing (excluding Twitter, blogs, and other informal variants of writing) all particles must be kept.
If you are unsure as to whether to keep or omit particles, a good rule of thumb is to err on the side of keeping the particles.
Can multiple particles be dropped in a single sentence?
これあげるよ。（＝これをあげるよ。） You can have this. (lit. "I'll give this [to you].")
おまえ昨日、学校行った？（＝おまえは昨日、学校に行ったか。） Did you go to school yesterday? (slightly masculine)
あたしスイカ好き。（＝あたしはスイカが好き。） I like watermelon. (feminine)
が and を, as you recall, are dropped when the part of the sentence they mark is made into the scope/topic (marked by は):
図書館でこの本を借りました。 I borrowed this book at the library.
この本をは図書館で借りました。 (making この本を the scope)
この本は図書館で借りました。 (replaced ungrammatical をは with は)