Learn Hardcore Polish: Ta aplikacja pokazuje mniej więcej, na jakim poziomie jestem. - This app shows more or less what level I am at.

aplikacja ends in -a, which is very often a marker of feminine gender.
The demonstrative pronoun this must agree with the noun in gender, number and case.

So:

ta aplikacja = this application (feminine)
ten telefon = this phone (masculine)
to mieszkanie = this flat (neuter)

Using ten aplikacja would mix masculine pronoun with a feminine noun, which is ungrammatical.

What does pokazuje mean exactly, and why not pokazuję?

The verb pokazywać means to show (imperfective). Its present-tense forms are:

(ja) pokazuję – I show / I am showing
(ty) pokazujesz – you show
(on/ona/ono) pokazuje – he/she/it shows
(my) pokazujemy – we show
(wy) pokazujecie – you (pl.) show
(oni/one) pokazują – they show

In the sentence:

Ta aplikacja pokazuje…
Literally: This application shows…
Subject: aplikacja (third person singular) → verb must also be third person singular: pokazuje, not pokazuję (which is first person singular).

Why is there mniej więcej here? Could we just say Ta aplikacja pokazuje, na jakim poziomie jestem?

mniej więcej means more or less / roughly / approximately.

Ta aplikacja pokazuje, na jakim poziomie jestem.
→ This app shows what level I’m at. (sounds more exact)
Ta aplikacja pokazuje mniej więcej, na jakim poziomie jestem.
→ This app shows more or less what level I’m at. (approximate, not exact)

Yes, you can drop mniej więcej. You’ll then sound like the app gives a more precise assessment. With mniej więcej, you’re clearly saying it’s only an estimate.

Where can mniej więcej go in the sentence? Is the comma after it necessary?

Typical and natural positions:

Ta aplikacja mniej więcej pokazuje, na jakim poziomie jestem.
Ta aplikacja pokazuje mniej więcej, na jakim poziomie jestem.

Both are fine and mean the same. In speech, it often comes right after the verb.

About the comma:

Polish usually puts a comma before a subordinate clause, which often starts with words like że, żeby, ponieważ, kiedy, na jakim… etc.
Here: na jakim poziomie jestem is a subordinate clause (it answers what? after pokazuje).

So:

Ta aplikacja pokazuje mniej więcej, na jakim poziomie jestem.
The comma is standard and expected in correct Polish spelling.

Why is it na jakim poziomie and not something like jaki poziom?

Both structures are possible, but they differ slightly:

na jakim poziomie jestem
- literally: on what level I am
- na
  - poziomie (locative case) = on (what) level
- very idiomatic when talking about skill/ability levels: na jakim poziomie jestem z polskiego? (what level am I at in Polish?)
jaki mam poziom
- literally: what level I have
- also possible: Ta aplikacja pokazuje mniej więcej, jaki mam poziom.
- similar meaning, maybe feels slightly more casual and less “structured” than na jakim poziomie jestem.

So na jakim poziomie sounds very natural when talking about CEFR levels, language levels, etc.

Why is poziomie in that form? What case is it?

poziomie is the locative singular of poziom (level).

For masculine nouns like poziom:

nominative (dictionary form): poziom
locative singular: poziomie

The preposition na in Polish can take:

accusative → movement onto something
- kładę książkę na stół – I put the book onto the table
locative → being on something / in some state/level
- książka leży na stole – the book is on the table
- jestem na wysokim poziomie – I am at a high level

In na jakim poziomie jestem, we are talking about a state/level, not movement → so na + locative: na jakim poziomie.

Could I say na jakim jestem poziomie instead of na jakim poziomie jestem?

Yes, you can. Polish word order is flexible, especially in subordinate clauses.

na jakim poziomie jestem
na jakim jestem poziomie

Both are grammatically correct and mean the same: what level I’m at.

The first option (na jakim poziomie jestem) is more common and sounds very natural and neutral. The second might sound a little more emphatic on jestem or stylistically different, but it’s still correct.

Why is there jestem at the end of the clause? In English we say “what level I am at”.

Polish doesn’t need to follow English word order. In subordinate clauses like this, it’s very natural in Polish to put the verb at the end:

… na jakim poziomie jestem.
… gdzie mieszkasz. (where you live)
… co lubisz. (what you like)

So:

English: what level I am at
Polish: na jakim poziomie jestem
literally: on what level am (I)

The subject “I” is built into the verb ending -em in jestem, so no separate pronoun is needed (ja is often omitted).

Why is jestem used and not something like mam (I have) or znajduję się (I am located)?

Different verbs give slightly different nuances:

jestem – I am
na jakim poziomie jestem – neutral, standard way to say what level I’m at.
mam – I have
jaki mam poziom – what level I have; also natural, a bit more direct/colloquial.
znajduję się – I am located / I am situated
na jakim poziomie się znajduję – grammatically OK, but sounds more formal or overcomplicated for this context.

In everyday speech about skills, exams, CEFR levels, etc., być (to be) is the most natural choice:
na jakim poziomie jestem.

Why is there no się with pokazuje? In English we might say “shows itself” or “is shown”.

pokazywać is a normal transitive verb: it takes a direct object and does not need się here.

Ta aplikacja pokazuje (co?) → na jakim poziomie jestem.
This app shows (what?) what level I’m at.

You only use się with pokazywać in reflexive or passive-like expressions, e.g.:

pokazywać się – to show oneself / to appear
- On nie lubi się pokazywać. – He doesn’t like to show himself / be seen.

But that’s a different meaning. In your sentence, pokazuje is just shows, no reflexive needed.