I'm pretty sure I've seen verbs with no conjugations in some sentences, but to me, that doesn't seem like it should be possible, given their dictionary definitions. So what situations, if any, are verbs used by themselves in (like, just a plain 歩く within a sentence for example) and what do they mean in that context? (sorry if this is a really dumb question)
Not a dumb question at all! There are some grammar structures that require verbs to be in their dictionary form. For example, adding の or こと after a casual verb turns it into a noun (for 歩く, to walk turns into the act of walking).
Example: 歩くのはいい運動です。(Walking is good exercise.)
It's the same with the "before" and "after" clauses as well.
Example: 歩く前に、ストレッチをします。(Before I walk, I stretch.)
There's plenty of other grammar points that use the casual form as well. It's not uncommon to drop the conjugation in casual speech either, especially if you're well acquainted with the person you're talking to.
A lot of the time you can use dictionary-form verbs to ask casual questions. For example, 行くの？ ("are you going?") or 立てる？ ("can you stand?"). Even just saying やる？ can mean "wanna do it?". In the same vein, these forms are also used for casual offers or invitations - 欲しい？ when you're offering someone something, or 食べる？ when you're offering food. Note that your intonation is very important here.
(As already mentioned above) You also see them with nominalisation, for example 歩くのが好き. Here, の turns the verb into a noun phrase: "the act of walking".
They also appear in expressions like 本を読む時間, where the verb is describing the noun - for example, 寝る時間 ("time to sleep").
Dictionary-form verbs are also used when you're describing something habitual, such as 毎朝コーヒーを飲む.
They also show up in conditional chains, like ボタンを押すと ... ("when you push the button... X happens"), which is very common in explanations.
Edit: 欲しい？ is an adjective, but it's still a neat pattern... I'm a bit tired, sorry :D
