Learn Hardcore Japanese: syuumatu ni ziyuuna zikan ga arimasu.

The particle に marks a specific point in time (or starting point) when something exists or happens. With time words like 週末, 金曜日 or 明日, you use に to mean “on the weekend,” “on Friday,” or “tomorrow at (a specific hour).”

What role does が play in 時間が? Why not use を or は?

Here が is the subject marker for existence verbs like ある／います.

時間がある literally means “time exists,” which we understand as “to have time.”
You cannot use を because ある is intransitive (it doesn’t take a direct object).
You could say 週末は自由な時間があります to topicalize “weekends,” but that changes the nuance: は highlights contrast or topic, whereas が simply marks “time” as the thing that exists.

Why is 自由な used before 時間? Could I say 自由時間 or 自由の時間?

自由 is a な-adjective, so when it directly modifies a noun you attach な: 自由な時間 (“free time”).

自由時間 (no な) isn’t standard; learners sometimes see compounds like 正午前 but 自由な時間 is required here.
自由の時間 is ungrammatical because 自由 isn’t a noun that takes の to modify “time.”

Why do we use あります here? Could we use います or 持っています instead?

あります is the polite inanimate-existence verb. We use it for things that are not alive (時間 is inanimate).
います is for animate beings (people, animals).
持っています (to hold/possess) is possible but less natural: Japanese usually say 時間がある rather than 時間を持っている.

Why is there no subject like 私 in the sentence? Where would it go if I wanted to include it?

Japanese often omits pronouns when context is clear. If you want to specify I, you can add 私は at the start:
「私は週末に自由な時間があります。」
But this sounds a bit stiff, because having free time is assumed about yourself.

Could I say 週末は自由な時間があります instead of using に?

Yes, but the nuance shifts:

週末に自由な時間があります simply states “I have free time on weekends.”
週末は自由な時間があります sets “weekends” as the topic and may imply contrast (“As for weekends, (unlike weekdays,) I have free time”).

Does the word order matter? Can I say 自由な時間が週末にあります or 時間が週末に自由ながあります?

Japanese is relatively flexible, but the most natural order is:
[time] に → [describing adjective + noun] → が → [existence verb]
So 週末に自由な時間があります is the preferred sequence.
Putting 自由な時間が週末にあります is grammatically okay but less common—time usually comes first. The last example 時間が週末に自由ながあります is ungrammatical because 自由な must directly precede 時間.

Can I omit に in casual speech, saying 週末自由な時間があります?

Dropping に makes the sentence feel clipped or slangy and can cause confusion. It’s best to keep に for clear, correct Japanese, even in casual contexts.