Learn Hardcore German: Meine Tante zahlt die Tickets ausschließlich online. - My aunt pays for the tickets exclusively online.

It’s accusative plural. The verb pays takes a direct object, so Tickets is in the accusative. The plural definite article is die for both nominative and accusative. Singular would be das Ticket (accusative: das Ticket); plural is die Tickets (accusative: die Tickets). The dative plural would be den Tickets (e.g., mit den Tickets).

Why is it die Tickets when Ticket is neuter?

Because in the plural, all genders use die in the nominative and accusative. Neuter only shows as das in the singular.

Is zahlt die Tickets idiomatic, or should I use bezahlt?

Both are acceptable. Nuance:

bezahlen + direct object is always safe: Meine Tante bezahlt die Tickets.
zahlen is commonly used with an amount or a bill: den Preis/Rechnung/Miete/Steuern zahlen; and with für + accusative: für die Tickets zahlen.
Many speakers also say die Tickets zahlen; others prefer die Tickets bezahlen. You won’t be misunderstood either way.

Could I say Meine Tante zahlt für die Tickets ausschließlich online?

Yes. That’s perfectly natural with zahlen + für + Akkusativ. Meaning and emphasis stay essentially the same.

What is the function and placement of ausschließlich online?

It’s an adverbial of manner/medium (how/where digitally she pays). Neutral word order puts it after the direct object: … zahlt die Tickets ausschließlich online. You can also front it for emphasis: Ausschließlich online zahlt meine Tante die Tickets.

Can I just use nur instead of ausschließlich? Are they identical?

nur is the everyday, lighter word for only; ausschließlich is stronger and a bit more formal/explicit (exclusively). In most contexts, nur online and ausschließlich online communicate the same idea; ausschließlich removes ambiguity more clearly. Negation differs:

nicht nur online = not only online (she also uses other ways)
nicht ausschließlich online = not exclusively online (same idea, but more formal)

Why is ausschließlich before online and not after it?

ausschließlich restricts or focuses what follows. It naturally precedes the element whose scope it limits: ausschließlich online (and not offline). Placing it after online is unusual in standard German.

What part of speech is online, and how is it written?

online is an adverb here (like digitally/via the internet). It’s written in lower case in the middle of a sentence: … ausschließlich online. It would only be capitalized at the start of a sentence: Online zahlt …

Why are Meine and Tante capitalized?

Meine is capitalized because it’s the first word of the sentence (otherwise it’s lowercase: meine).
Tante is a noun; all German nouns are capitalized.

What tense and person is zahlt?

3rd person singular, present tense (Präsens) of zahlen. Quick forms: ich zahle, du zahlst, er/sie/es zahlt, wir zahlen, ihr zahlt, sie/Sie zahlen. bezahlen conjugates the same way (ich bezahle, …; er/sie/es bezahlt).

How do I say this in the past?

Use Perfekt:

Meine Tante hat die Tickets ausschließlich online gezahlt.
Meine Tante hat die Tickets ausschließlich online bezahlt. Both are fine. Participle forms: gezahlt, bezahlt.

Can I move elements around for emphasis?

Yes. German allows fronting for focus:

Die Tickets zahlt meine Tante ausschließlich online. (focus on the tickets)
Ausschließlich online zahlt meine Tante die Tickets. (focus on the exclusivity/online) The original is the neutral order. Avoid splitting ausschließlich from online.

How would I negate this properly?

She does not pay online: Meine Tante zahlt die Tickets nicht online.
She does not pay exclusively online (i.e., also in other ways): Meine Tante zahlt die Tickets nicht ausschließlich online.

Is online the only way to express this idea?

No. Common alternatives:

… ausschließlich im Internet.
… ausschließlich über das Internet / übers Internet.
… ausschließlich per Online-Zahlung (more formal/technical). online is the most concise and idiomatic in everyday German.

Is Tickets the best word here? What about Karten, Fahrkarten, Eintrittskarten?

Ticket is widely used in modern German and is fine. More specific:

Fahrkarten for transport tickets.
Eintrittskarten for event/entry tickets.
Karten is a general word for tickets (context makes meaning clear). All would fit the sentence with the appropriate article and plural: die Fahrkarten, die Eintrittskarten, die Karten.