The question of transitivity can get tricky, especially when discussing and using languages as different as English and Japanese.
First, some background.
Kinds of transitivity
Most English grammars appear to discuss "transitivity" in terms of the syntax or wording of a sentence. A verb is deemed to be "transitive" if it is followed by a direct object.
- I eat an apple. → Transitive, as the verb "eat" is followed by the direct object "an apple".
- I eat. → Intransitive, as there is no direct object explicitly included in this sentence.
Most Japanese grammars appear to discuss this concept differently, where the transitivity of a verb is considered in terms of the semantics or meaning of the word. The Japanese terms for the verb types are 他動詞 (literally "other-moving / other-acting word") and 自動詞 (literally "self-moving / self-acting word"). A verb is deemed to be a 他動詞 if the action of the verb involves the actor (the "subject" of the verb, in English grammar terms) performing some action that affects another noun (the "object" of the verb). Importantly, this applies regardless of whether that object is explicitly included in the sentence.
- 私はリンゴを食べる。 → Transitive in syntactic terms, as the verb is preceded by the direct object リンゴを. Also transitive in semantic terms, as the semantics (meaning) of the verb 食べる requires the existence of an object: one cannot "eat" without eating something.
- 私は食べる。 → Intransitive in syntactic terms, as there is no direct object explicitly included in this sentence. However, this is still transitive in semantic terms: the verb 食べる is a 他動詞, even when the object is left unstated.
Now that we've established a framework for discussion, let's look again at your questions.
Is it possible for a single verb in Japanese to have both transitive and intransitive uses?
In terms of syntax, Japanese is famous for the degree to which various portions of a sentence can be omitted. As such, pretty much any 他動詞 verb in Japanese can be used syntactically transitively and syntactically intransitively. Some 自動詞 verbs in Japanese can also be used both syntactically transitively and syntactically intransitively (more on that below).
In terms of semantics, single verbs that have both transitive and intransitive senses were much more common in Old Japanese; over time, different conjugated forms became more widespread to express one or the other. Take 付く, for instance. This was both a 自動詞 and a 他動詞 in older Japanese, but now, the root form 付く is 自動詞. The 他動詞 senses historically used the 下二段活用 (lower bigrade conjugation) pattern that became the -eru ending in 付ける. Another example is 広ぐ. The root form is no longer used, while the 自動詞 senses are expressed by 広がる, and the 他動詞 senses by 広げる.
There are still a few verbs in modern Japanese that can be used as either 自動詞 or 他動詞. Most of the ones I'm aware of are Sino-Japanese (kanji terms read with on'yomi and followed by する), such as 完成する.
If a verb is marked transitive in a dictionary, is that a guarantee that the verb has a specific object (that would be marked with を, か, or と) somewhere within the context of the conversation (it may be unstated)?
I assume that you're describing Japanese-to-English dictionaries. I believe these all base their "intransitive / transitive" descriptions on the 自動詞・他動詞 distinctions in Japanese. If my understanding is correct, then yes, a "transitive verb" in this case would have an object, either explicitly included in the sentence or left unstated and implied (as in the "eat / 食べる" example above).
"Direct objects" in Japanese
Note that the presence of a direct object in Japanese (a noun marked with を) does not necessarily indicate that the verb is a 他動詞. Consider the sentence, 私は道を歩く。 This is literally "I walk the road." Syntactically, this could be considered a transitive use of the verb 歩く "walk". However, semantically, the action of 歩く-ing does not affect the road in any way -- it affects the actor, 私, instead: so 歩く is still a 自動詞, even though this sentence includes a direct object.
This is similar to how 向く works, as described in snailplane's answer to your other question.