Αν είχα περισσότερη όρεξη, θα μπορούσα να τελειώσω όλη την εργασία σήμερα.

Αν means if. In modern spoken Greek, αν and εάν are essentially the same in meaning.

• αν = more common, more colloquial
• εάν = a bit more formal / written style

You can say:

• Αν είχα περισσότερη όρεξη, ...
• Εάν είχα περισσότερη όρεξη, ...

Both are correct and mean the same thing in everyday use.

Είχα is the imperfect of the verb έχω (to have).

• έχω = I have (present)
• είχα = I had (imperfect / past)

In conditionals like this, Greek usually uses a past tense (often the imperfect) in the αν-clause to express an unreal/hypothetical present situation.

So:

• Αν έχω περισσότερη όρεξη = If I have more energy (real, open possibility)
• Αν είχα περισσότερη όρεξη = If I had more energy (but I don’t; it’s hypothetical)

This is a hypothetical / unreal present conditional.

Structure here:

• Αν είχα περισσότερη όρεξη
  → αν

  • past tense (imperfect) in the indicative (είχα)

• θα μπορούσα να τελειώσω...
  → θα

  • past (here imperfect of μπορώ → μπορούσα), often called the “conditional” form in practice

Meaning: a situation that is not true right now, but imagined:

If I had more energy (but I don’t), I could finish all the work today.

In Greek, when a sentence starts with a dependent clause (like an αν-clause), it is usually followed by a comma:

• Αν είχα περισσότερη όρεξη, θα μπορούσα...

If you reverse the order, the comma is usually omitted:

• Θα μπορούσα να τελειώσω όλη την εργασία σήμερα αν είχα περισσότερη όρεξη.

So the comma is there mainly for clarity and rhythm, not because the meaning changes.

Literally, όρεξη means appetite (for food).
But in everyday Greek it also often means:

• motivation
• willingness
• “mood to do something”

Here, περισσότερη όρεξη is best understood as:

• more energy / motivation / willingness to work

So even though it literally could be “more appetite”, in this context it’s about having the drive to finish the work.

Both are possible and common:

• περισσότερη όρεξη
• πιο πολλή όρεξη

They both mean more appetite/energy/motivation.

Nuance:

• περισσότερη = a single comparative form of πολύς (“much/many”), and sounds a bit more neutral/standard.
• πιο πολλή = πιο (more) + πολλή, also very common and natural in speech.

In this sentence, περισσότερη όρεξη is perhaps a bit more compact and slightly more “careful” style, but there is no real difference in meaning.

Περισσότερη is:

• Gender: feminine
• Number: singular
• Case: accusative
• Degree: comparative (“more”)

It agrees with όρεξη, which is:

• η όρεξη (feminine, singular, accusative here).

So:

• περισσότερη (fem. sg. acc.) όρεξη (fem. sg. acc.)
  They match in gender, number, and case, as adjective and noun.

Each form has a different meaning:

• μπορούσα να τελειώσω = I could / was able to finish (in the past, real situation)
• θα μπορώ να τελειώσω = I will be able to finish (future, real expectation)
• θα μπορούσα να τελειώσω = I could / would be able to finish (hypothetical, unreal)

Here we talk about a hypothetical ability that depends on a condition that is not met, so Greek uses:

• θα μπορούσα = would/could (conditional sense)

That’s why θα μπορούσα fits the unreal conditional meaning:
“If I had more energy, I could/would be able to finish all the work today.”

Να here introduces the subjunctive form of the verb:

• να τελειώσω = (to) finish

After verbs of ability, desire, intention, etc. (μπορώ, θέλω, πρέπει, etc.), Greek commonly uses:

• verb + να + subjunctive

So:

• μπορώ να τελειώσω = I can finish
• θα μπορούσα να τελειώσω = I could / would be able to finish

Without να, τελειώσω could not appear in this form; να is needed to mark the subjunctive clause.

Greek distinguishes aspect (completed vs ongoing) in the subjunctive:

• να τελειώσω (aorist subjunctive) = to finish (as a single, completed action)
• να τελειώνω (present subjunctive) = to be finishing / to keep finishing (ongoing / repeated)

Here the idea is: finish all the work completely today. That’s a single completed action, so Greek naturally uses the aorist subjunctive:

• να τελειώσω όλη την εργασία σήμερα = to finish all the work today (get it done).

Όλη την εργασία is in the accusative case, because it is the direct object of τελειώσω:

• να τελειώσω (τι;) όλη την εργασία
  “to finish (what?) all the work”

Breakdown:

• όλη = feminine, singular, accusative of όλος (“all”)
• την = feminine, singular, accusative definite article
• εργασία = feminine noun; here also in accusative (same form as nominative in sg., but functionally accusative)

So the phrase is “all the work” as the thing being finished.

Yes, you can also say:

• Αν είχα περισσότερη όρεξη, θα μπορούσα να τελειώσω όλη τη δουλειά σήμερα.

Differences:

• εργασία = work in a more formal / neutral / often academic or professional sense; also “assignment”, “paper”, “project”.
• δουλειά = more colloquial, everyday “work / job / stuff to do”.

In a student context, εργασία can mean a specific assignment:

• όλη την εργασία = all of the assignment / all the work that has been given

In everyday work context, δουλειά is more common:

• έχω πολλή δουλειά = I have a lot of work.

Yes, you can switch the clauses:

• Αν είχα περισσότερη όρεξη, θα μπορούσα να τελειώσω όλη την εργασία σήμερα.
• Θα μπορούσα να τελειώσω όλη την εργασία σήμερα αν είχα περισσότερη όρεξη.

Both mean exactly the same. The only differences are:

• Punctuation: comma after the αν-clause only when it comes first.
• Emphasis:
  
  • Starting with Αν είχα περισσότερη όρεξη slightly emphasizes the condition.
  • Starting with Θα μπορούσα να τελειώσω... slightly emphasizes the result.

Yes, that sentence is also correct:

• Αν είχα περισσότερη όρεξη, θα τελείωνα όλη την εργασία σήμερα.

Comparison:

1. θα μπορούσα να τελειώσω

   • Focus on ability / possibility: “I would be able to finish.”

2. θα τελείωνα (from τελειώνω, imperfect with θα)

   • Focus on the action itself: “I would finish / I’d be finishing (it).”

Both are unreal conditionals about now/today.
In practice, they are very close in meaning; the original with θα μπορούσα να τελειώσω sounds slightly more like:

I could finish it (if I had more energy),

whereas θα τελείωνα sounds more like:

I would (actually) finish it (if I had more energy).

In Greek, time adverbs like σήμερα are quite flexible. All of these are acceptable:

• Αν είχα περισσότερη όρεξη, σήμερα θα μπορούσα να τελειώσω όλη την εργασία.
• Αν είχα περισσότερη όρεξη, θα μπορούσα σήμερα να τελειώσω όλη την εργασία.
• Αν είχα περισσότερη όρεξη, θα μπορούσα να τελειώσω σήμερα όλη την εργασία. (the original)
• Σήμερα, αν είχα περισσότερη όρεξη, θα μπορούσα να τελειώσω όλη την εργασία.

The basic meaning doesn’t change: it refers to today.
Different positions can add slight rhythmic or emphasis differences, but all are natural. The original …να τελειώσω όλη την εργασία σήμερα is very typical word order.