How important is learning keigo (敬語)? Is it necessary to master?
How often you will need to use keigo depends on what situations you use Japanese in. If you plan on doing a job search in Japanese, working in Japan, or communicating with professionals in Japanese, at least basic proficiency with speaking keigo is recommended. It shows both respect and proficiency in the language. Keigo is also useful when living in Japan, as clerks in shops or waiters in restaurants typically are expected to speak to customers with keigo. In this case, you should be familiar with understanding it, but it is not necessary to be able to respond with keigo.
What is the difference between the particles は and が?
は is known as a "topic marker," while が is a "subject marker." What this means is that は is used to mark the topic of a sentence, which provides context for the sentence (e.g. In「車はどこですか」, は helps to provide the context of what is being asked about), while が distinguishes something as the subject of a verb or provides distinction between many choices (e.g. Answering 「私が車を運転しました」 when asked who drove, since が distinguishes you as the one who did the action).
What does the grammar ～あげく mean? How do I use it with either verbs or nouns?
～あげく is a grammar point that is used to express that a result (typically negative) was attained after a great deal of effort. Because it is used to express results, it should be used to talk about past actions. To modify verbs, attach ～あげく to the end of the verb's past tense (e.g. 勉強したあげく). To modify nouns, put the particle の after the noun and before ～あげく (e.g. 相談のあげく).