Learn Hardcore Turkish: Ben epey yoruldum.

You can drop it. Turkish verb endings already show the subject, so Epey yoruldum is the most neutral, natural version. Using Ben adds contrast/emphasis, like “I (as opposed to others) got quite tired,” or it helps answer a who-focused question.

What exactly does epey mean, and how is it different from çok, oldukça, hayli, bayağı, and epeyce?

All of these roughly mean “quite/very/pretty,” but their flavor and register vary:

çok: very/a lot; the most common, neutral intensifier.
epey: quite/considerably; slightly softer than çok, natural in speech and writing.
oldukça: rather/quite; a bit more formal or careful.
hayli / bir hayli: quite a bit; somewhat formal/literary but common in writing.
bayağı: pretty/quite (colloquial); note that as an adjective it can mean “vulgar,” but as an adverb it’s fine.
epeyce: a variant of epey; often a touch more colloquial or emphatic.

All can modify verbs or adjectives; epey and epeyce can also go before nouns (e.g., epey para).

Where does epey go in the sentence? Is Ben epey yoruldum the only possible order?

Adverbs like epey usually come right before the verb or adjective they modify.

Most neutral: Epey yoruldum.
With subject focus: Ben epey yoruldum.
Sentence-initial for overall emphasis: Epey yoruldum ben. (afterthought style; colloquial) Putting epey after the verb (Yoruldum epey) is generally odd in standard speech.

Why yoruldum and not yorgunum?

yoruldum uses the verb yorulmak (“to get tired”) in simple past and highlights the process/result: you became tired through activity; it often implies “and I’m tired now.”
yorgunum uses the adjective yorgun (“tired”) with the copula and simply states your current state, without pointing to the tiring process. Both are common; after exertion, yoruldum feels more natural, while in a status update, yorgunum may fit better.

How is yoruldum formed? Why -dum?

Morphology: yorul- (verb stem “get tired”) + -du (simple past) + -m (1sg) → yoruldum.

Vowel harmony picks -du (not -dı/-di/-dü) because the last vowel in yorul- is back and rounded (u).
The past suffix surfaces as d or t depending on the preceding consonant’s voicing. Since l is voiced, you get -du (not -tu).

How do I say “I’m getting tired right now” or “I get tired easily”?

Ongoing process: yoruluyorum (“I’m getting tired”).
Habitual/general tendency: yorulurum (“I get tired [easily/usually]”).
Add an adverb for clarity: çabuk yoruluyorum (“I get tired quickly/easily”).

Is this sentence formal or informal?

Neutral. You can use it almost anywhere. In very formal or delicate contexts you might soften it with context, e.g., Bugün epey yoruldum; kısa bir ara versek uygun olur mu?

How do I pronounce it? Where is the stress?

Ben: [ben]
epey: [eˈpej] (stress on the second syllable: e-PEY; the final -ey sounds like English “ay”)
yoruldum: [joɾulˈdum] (yo-rul-DUM; final stress; r is a tapped/flapped r; u is like “oo” in “food”)

Is there any difference between Ben epey yoruldum and Epey yoruldum?

Meaning is the same, but Ben adds contrast/focus on “I.” Without Ben is the default, smoother option unless you need emphasis.

How do I make it negative or ask it as a question?

Negative: Epey yorulmadım (“I didn’t get very tired”).
Yes–no question: Epey yoruldun mu? / Yoruldun mu?
To ask about state (not the process): Yorgun musun? Negative of the adjective: Yorgun değilim.

Can epey be used with nouns too?

Yes:

epey para (quite a bit of money)
epey bir süre (quite a while)
epeyce kişi (quite a few people) The pattern epey bir + noun is common in speech.

Is epeyce correct? Any difference from epey?

Yes, epeyce is correct. It usually feels slightly more colloquial or emphatic. In this sentence, Epeyce yoruldum is perfectly acceptable and close in meaning to Epey yoruldum.

Is bayağı okay to use here?

Yes: Bayağı yoruldum is a natural colloquial alternative. Just remember bayağı also means “vulgar/lowbrow” when used as an adjective about people/things, but that doesn’t affect its adverbial use here.

Can I say çok fazla yoruldum? What about fazla yoruldum?

Çok yoruldum: very tired (most common strong intensifier).
Fazla yoruldum: I got “too” tired (more than desirable).
Çok fazla yoruldum: colloquial for “way too tired”; sounds like strong complaint. Use when you really want to emphasize excess.

What’s the difference between yoruldum and yorulmuşum?

yoruldum: plain past; you directly assert the fact.
yorulmuşum: evidential past; you realized or are reporting it indirectly (“apparently/seems I got tired”), often softer or self-reflective: Epey yorulmuşum.