Ta roślina doniczkowa nie ma ani mszyc, ani przędziorków.

Ta means this. It is the feminine singular form, because roślina is a feminine noun.

So:

• ten = this for masculine nouns
• ta = this for feminine nouns
• to = this for neuter nouns

Since roślina is feminine, you say ta roślina.

Ma is the he/she/it has form of mieć, which means to have.

So:

• ma = has

In this sentence, the subject is ta roślina doniczkowa, so ma means has:

• Ta roślina ma... = This potted plant has...
• Ta roślina nie ma... = This potted plant does not have...

Doniczkowa is an adjective meaning something like potted or for pots / pot-grown. It agrees with roślina in gender, number, and case.

Because roślina is feminine singular nominative, the adjective must also be feminine singular nominative:

• roślina
  • doniczkowa

This is standard adjective agreement in Polish.

Polish usually forms simple negation by putting nie before the verb.

So:

• ma = has
• nie ma = does not have

This is completely normal and very common. In fact, nie ma is one of the most common negative patterns in Polish.

Ani ... ani ... means neither ... nor ... or, in a sentence like this, not ... or ... either.

So:

• nie ma ani mszyc, ani przędziorków = it has neither aphids nor spider mites

A very important point: ani normally appears with a negative verb, here nie ma.
So the structure is:

• nie
  • verb + ani X, ani Y

Because after nie ma, Polish uses the genitive.

The basic forms are:

• mszyca = an aphid
• mszyce = aphids
• przędziorek = a spider mite
• przędziorki = spider mites

But after nie ma, they change to genitive plural:

• mszyc
• przędziorków

So this sentence is a good example of a very common pattern:

• affirmative possession often uses accusative
• negative nie ma uses genitive

Yes. That is also natural Polish.

These are both possible:

• nie ma ani mszyc, ani przędziorków
• nie ma mszyc ani przędziorków

The version with ani ... ani ... is a little more explicitly balanced, like neither ... nor ... in English. The shorter version is also very common in everyday speech.

Because the sentence is about one plant, but it says that this one plant does not have any aphids or any spider mites.

That is perfectly normal:

• one subject: ta roślina
• plural things it does not have: mszyc, przędziorków

English works the same way:

• This plant doesn’t have aphids or spider mites.

Yes, very natural. Roślina doniczkowa is the normal term for a houseplant or potted plant.

Depending on context, Polish could also say something like roślina w doniczce, but roślina doniczkowa is a standard dictionary-style expression and sounds natural.

It is in the nominative, because it is the subject of the sentence.

You can see the basic subject pattern here:

• Ta roślina doniczkowa = subject
• nie ma = verb
• ani mszyc, ani przędziorków = object/complement in the genitive because of nie ma

So the sentence structure is very typical for Polish.

Polish does not have articles like English the and a/an.

So roślina can mean:

• a plant
• the plant

The exact meaning depends on context. In this sentence, ta already makes it specific:

• ta roślina = this plant

So no article is needed.

A rough guide:

• ta ≈ tah
• roślina ≈ rosh-LEE-na
• doniczkowa ≈ doh-neech-KO-va
• nie ma ≈ nyeh mah
• ani ≈ AH-nee
• mszyc ≈ something like mshits
• przędziorków ≈ roughly pshen-JOR-koof

A few helpful pronunciation notes:

• ś is a soft sh sound
• cz sounds like English ch
• rz and ż are usually like the s in measure
• Polish stress is usually on the second-to-last syllable

So:

• rośLIna
• doniczKOwa
• przędziorKÓW