Learn Hardcore Greek: Η βαλίτσα της έχει πολλά πράγματα.

Many, but not all, feminine nouns end in -α (or -η).
Many masculine nouns end in -ος, but -ος can also be feminine or neuter, so the article is important.

So η βαλίτσα (with η) tells you clearly that this noun is feminine. The -α ending is a strong hint, but you should always learn the noun together with its article (η/ο/το).

What exactly is της here? Why is it “her” and not “she”?

Της here is a weak genitive pronoun used as a possessive:

μου – my
σου – your (singular)
του – his / its (masc or neuter)
της – her / its (feminine)
μας – our
σας – your (plural/formal)
τους – their

In η βαλίτσα της:

βαλίτσα = suitcase
της = her

So η βαλίτσα της = her suitcase.

It cannot mean she; she as a subject is αυτή in Greek.
Της is the “of her” / her form, not the subject form.

Why is της after the noun (βαλίτσα) and not before, like English “her suitcase”?

In Greek, possessive pronouns of this type normally go after the noun:

η βαλίτσα της – her suitcase
το σπίτι μου – my house
το αυτοκίνητό του – his car

So the natural order is:
article + noun + possessive pronoun.

If you want strong emphasis on her, you can say:

η δική της βαλίτσα – her (own) suitcase, stressing that it’s hers (not someone else’s).

But you do not say η της βαλίτσα with just της before the noun; that’s not how modern Greek normally forms this type of possessive.

What form of the verb is έχει and how would I say “I have”, “you have”, etc.?

Έχει is the 3rd person singular, present tense of the verb έχω (to have):

έχω – I have
έχεις – you have (singular)
έχει – he / she / it has
έχουμε – we have
έχετε – you have (plural / formal)
έχουν(ε) – they have

So in the sentence:

η βαλίτσα της έχει – her suitcase has (many things).

Does έχει here literally mean “has”, or is it more like “there are many things in her suitcase”?

Grammatically it is “has”, but in context it usually means “contains / there are many things in it”.

Η βαλίτσα της έχει πολλά πράγματα.
= Her suitcase has many things.
= There are many things in her suitcase.

Both English versions express the same idea; Greek uses έχει very naturally in this way.

What is the difference between πολύ, πολλά, πολλοί, and πολλές?

They all come from the same word meaning much / many / a lot, but they change to agree with the noun:

πολύ – neuter singular, also adverb (very / a lot)
πολλά – neuter plural (many, for neuter plural nouns)
πολλοί – masculine plural (many, for masculine plural nouns)
πολλές – feminine plural (many, for feminine plural nouns)

Examples:

πολλά πράγματα – many things (neuter plural)
πολλοί άνθρωποι – many people (masculine plural)
πολλές καρέκλες – many chairs (feminine plural)

In πολλά πράγματα, πολλά matches the gender and number of πράγματα (neuter plural).

Why is there no article before πολλά πράγματα? Why not τα πολλά πράγματα?

Without an article, πολλά πράγματα means “many things / a lot of things” in a general, indefinite way.

πολλά πράγματα – many things (not specific, just quantity)
τα πολλά πράγματα – the many things (a specific group that is already known or defined in context)

In this sentence, we are just saying her suitcase contains a lot of stuff in general, so the indefinite form πολλά πράγματα (no article) is correct and natural.

Why is πολλά in this form? What is it agreeing with?

Πολλά is in the neuter plural accusative form, because:

πράγματα is neuter plural accusative (direct object of έχει)
Adjectives and quantifiers agree with the noun in gender, number, and case

So:

πολλά (neuter plural acc.)
πράγματα (neuter plural acc.)

They match, just like in English we say many things (plural-plural) rather than many thing.

Why is πράγματα in the accusative case? How can I tell?

Πράγματα is the direct object of the verb έχει (what does the suitcase have?), so it is in the accusative case.

Clues:

The subject: Η βαλίτσα της (nominative)
The verb: έχει
The direct object: πολλά πράγματα (accusative)

For neuter nouns, nominative and accusative plural look the same in form (πράγματα can be both), so you rely on sentence position and function:

Subject position → nominative
Direct object of a verb → accusative

Here, πράγματα clearly answers “has what?”, so it’s functioning as the object → accusative.

Can the word order change? For example, can I say Έχει πολλά πράγματα η βαλίτσα της?

Yes, Greek allows relatively flexible word order.

All of these are possible and grammatical:

Η βαλίτσα της έχει πολλά πράγματα. (neutral, very natural)
Η βαλίτσα της πολλά πράγματα έχει. (slightly marked, often for emphasis on πολλά πράγματα)
Έχει πολλά πράγματα η βαλίτσα της. (emphasis starts with έχει / πολλά πράγματα)

The most standard, textbook word order here is the original:

Η βαλίτσα της έχει πολλά πράγματα.

How would I say “his suitcase has many things” or “their suitcase has many things”?

Just change the possessive pronoun:

Η βαλίτσα του έχει πολλά πράγματα.
– His suitcase has many things.
Η βαλίτσα τους έχει πολλά πράγματα.
– Their suitcase has many things.

Pattern:

η βαλίτσα μου – my suitcase
η βαλίτσα σου – your suitcase (sing.)
η βαλίτσα του – his suitcase
η βαλίτσα της – her suitcase
η βαλίτσα μας – our suitcase
η βαλίτσα σας – your suitcase (pl./formal)
η βαλίτσα τους – their suitcase

What are the singular and plural forms of βαλίτσα and πράγμα?

Βαλίτσα (feminine):

Singular: η βαλίτσα – the suitcase
Plural: οι βαλίτσες – the suitcases

Πράγμα (neuter):

Singular: το πράγμα – the thing
Plural: τα πράγματα – the things

In the sentence you have the plural:

πολλά πράγματα – many things.