One of the tricks with the whole “Japanese has no spaces” thing is that with experience, you can learn to recognise where words start and end. In particular, kanji helps immensely in this. Usually a stand-alone kanji word will have hiragana particles and conjunctions on both sides, allowing you to visually distinguish it.

Here’s an example:
小林さんの猫は超可愛いです

You can see the 猫 sitting by itself in the middle of the sentence, yes? That’s what he means by “when the kanji is all alone.” (It’s not, in all fairness, the best of the examples, because 猫 tends to use its kun’yomi even when it’s NOT all alone, but you get my drift anyway.)

As to how you immediately know: experience. Once you’ve managed to learn a bunch of kanji and vocab, you start to be able to intuit which reading goes where much more easily.

Aye, that’s down to experience as well.

That said, when you’re trying to tell which kun’yomi the kanji in a verb uses, the hiragana suffix will help. For example, 出す is だす, while 出る is でる, never the other way around. As with all rules in Japanese, there’s exceptions – for example, 入れる (はいれる) and 入れる (いれる) have the same hiragana ending.

Context is your friend, always – noone’s ever going to suddenly jump out of a bush at you and shout “quick! What’s the on’yomi of 寂?”. You’re pretty much only going to be seeing kanji in the context of a sentence, or in a set phrase on a sign. For example of this kind of set phrase, 立入禁止 means “No entry”. You’d expect the 立入 to use on’yomi, but since it’s the sort of thing which appears on a sign, it’s been abbreviated for clarity – it’s actually 立ち入り = たちいり. The full phrase is たちいりきんし.

The main exception to this is proper nouns – particularly the names of people and places. Context won’t be able to tell you anything. In these cases, however, it’s perfectly acceptable to simply ask someone how to say it. Even native Japanese speakers can stumble over proper nouns.