Le client paie en espèces.

Yes:

• en liquide: very common and informal-neutral (France)
• en cash: informal, anglicism, widely understood
• au comptant / payer comptant: “pay outright/at once”; not necessarily cash (could be by card), just not in installments
• Canadian French: en argent comptant, payer comptant are very common

Both spellings are accepted for -ayer verbs before a silent ending:

• je paie / je paye
• tu paies / tu payes
• il/elle paie / paye
• ils/elles paient / payent

In modern usage, forms with i (paie, paies, paient) are more common in France, but both sets are correct. In the future and conditional, both je paierai and je payerai are accepted.

Le client paie en espèces is typically:

• [lə kljɑ̃ pɛ ɑn‿ɛspɛs]

Notes:

• paie = [pɛ]
• There is a liaison in en espèces: the final consonant surfaces as [n], so you hear [ɑn‿ɛspɛs], and the vowel of en is denasalized in the liaison context.

It can be either, depending on context. French often uses the definite article for general statements. For a clearly general idea, you can also use the plural:

• Les clients paient en espèces.

Use the feminine noun:

• La cliente paie en espèces. Plural forms:
• Mixed or all-male: Les clients
• All-female: Les clientes

Both mean “to pay,” but régler is a bit more formal or business-like and often used with bills/invoices:

• Le client règle la note/facture en espèces.

Neutral and most natural is after the verb:

• Le client paie en espèces. Fronting for emphasis is possible but marked/stylistic:
• En espèces, le client paie.