Learn Hardcore Japanese: watasi ha kurezittokaado de haraimasu.

You can drop it. Japanese usually omits pronouns when context makes them clear. In everyday use you’d simply say: クレジットカードで払います. Keeping 私は is fine if you want to set yourself as the topic explicitly or contrast with someone else.

What exactly does the particle は do here? How is it different from が?

は marks the topic: “As for me, …” It’s neutral and sets the frame of conversation.
が marks the grammatical subject and often emphasizes it.
If you want to say “I (and not someone else) will pay,” use 私が払います.
Neutral statement: 私はクレジットカードで払います.
Contrast/emphasis: 私が払います.

Why is で used after クレジットカード? Why not を or に?

で marks the means/instrument: “by/with.” So クレジットカードで = “by credit card.”
を marks the direct object. You don’t “pay the card”; you pay a fee/amount: 料金を払う.
に marks a recipient/target: “pay to [someone].” e.g., 店員に払う “pay the clerk.”
So the full pattern can be: 料金をクレジットカードで店員に払います.

There’s no を-object here—what are we paying?

It’s understood from context (the fee, bill, fare, etc.). If you want to be explicit, add it:

代金/料金/会費/家賃/税金をクレジットカードで払います.

What’s the difference between 払う and 支払う? And what about お支払い?

払う is the common everyday verb “to pay.”
支払う is more formal/literary or businesslike (contracts, bills).
お支払い is the polite noun form (“payment”) with an honorific prefix.
Examples:
Casual/polite verb: カードで払います.
Formal: クレジットカードで支払います.
Polite noun phrase: クレジットカードでお支払いします.

Why is it 払います and not 払いです? When do I use ます vs です?

ます attaches to verbs (払う → 払います).
です is the copula used with nouns/na-adjectives (学生です, 便利です).
You can’t say “払いです” to mean “pay.” Use 払います.

How do I ask “Can I pay by credit card?” naturally?

Several natural options:

クレジットカードは使えますか。 “Do you accept credit cards?” (most common)
クレジットカードで払えますか。 “Can I pay by credit card?”
クレジットカードで大丈夫ですか。 “Is credit card okay?”
Short and natural at a register: カードでいいですか。 / カードでお願いします。

How do I change the tense or make it negative?

Past: 払いました (I paid)
Negative: 払いません (I don’t/won’t pay)
Past negative: 払いませんでした
Plain forms: 払った / 払わない / 払わなかった
Potential (can): 払えます
For permission: クレジットカードで払ってもいいですか。
Note: 払ってください means “please pay (you)”—not what you say to a cashier.

Can I rearrange the word order?

Japanese is flexible, but the natural spot for a “means” phrase (Xで) is before the verb. Best: (私は) クレジットカードで払います.
Putting the instrument after the verb is odd. You can move elements earlier for focus, but keep …で + verb together near the end for naturalness.

How do I emphasize “I” will pay?

Use が:

私が払います。 “I’ll pay.” (implies volunteering/contrast) If you’re arguing who pays, this clearly claims responsibility.

Does で also mean “at” for places? Can I combine place and means?

Yes. で can mark location (“at”) and means (“by”). You can combine them:

コンビニでクレジットカードで払います。 “I’ll pay at the convenience store by credit card.”
In speech, a short pause helps: コンビニで、カードで払います。

Why is クレジットカード in katakana? Can I just say カード or クレカ?

Loanwords are written in katakana. クレジットカード is the full term; in conversation people often say:

カード (most common)
クレカ (casual abbreviation)
All take で: カードで払います, クレカで払います.

Is “クレジットカードを使って払います” okay?

It’s grammatical but sounds wordy and redundant. The natural way is simply:

クレジットカードで払います。
Use …を使って only if you really need to contrast the tool or describe a more complex action.