Na prezentacji mam dwadzieścia jeden slajdów, ale projektor nie działał od razu.

Both can be possible, but they emphasize slightly different things.

• na prezentacji often means at/during the presentation or for the presentation
• w prezentacji means inside the presentation itself as a file, document, or set of slides

So in this sentence, Na prezentacji mam dwadzieścia jeden slajdów sounds natural if the speaker is talking about their presentation as an event or what they are presenting.

Compare:

• Na prezentacji mam 21 slajdów. = In my presentation / For the presentation, I have 21 slides.
• W prezentacji jest 21 slajdów. = There are 21 slides in the presentation.

For learners, the easiest rule is:

• use na for events/occasions: na lekcji, na spotkaniu, na konferencji, na prezentacji
• use w for being inside something: w książce, w pliku, w prezentacji

Yes. Mam is the 1st person singular of mieć = to have.

Here it means something like:

• I have twenty-one slides in my presentation
• My presentation has twenty-one slides

Polish often uses mieć in places where English might also say there are or use the thing itself as the subject.

So:

• Mam dwadzieścia jeden slajdów = I have twenty-one slides
• Moja prezentacja ma dwadzieścia jeden slajdów = My presentation has twenty-one slides

All of these are natural, just with slightly different focus.

Because after many numbers in Polish, the noun changes form.

With slajd, the pattern is:

• jeden slajd = 1 slide
• dwa / trzy / cztery slajdy = 2 / 3 / 4 slides
• pięć slajdów = 5 slides
• dwadzieścia jeden slajdów = 21 slides
• dwadzieścia dwa slajdy = 22 slides
• dwadzieścia pięć slajdów = 25 slides

A useful shortcut:

• numbers ending in 2, 3, 4 usually take the slajdy type
• most other numbers take the slajdów type
• but 12, 13, 14 are exceptions and also take slajdów

So 21 takes slajdów, not slajdy.

Slajdów is genitive plural.

That may seem strange, because the whole phrase is the object of mam, so you might expect accusative. But Polish number phrases are special: after many numbers, the noun appears in the genitive plural.

So in:

• mam dwadzieścia jeden slajdów

the noun slajdów is in the form required by the numeral phrase.

This is one of the trickier parts of Polish grammar, so it is very normal to need time with it.

It is a borrowed word from English, but it is completely normal in modern Polish, especially for presentations.

Its common forms are:

• slajd = slide
• slajdy = slides
• slajdów = of slides / slides after certain numbers

So this sentence sounds natural and everyday. You will hear slajd all the time in office, school, and conference contexts.

Działał is the past tense, and it agrees with the subject projektor.

• projektor is masculine singular
• so the past tense is działał

Compare:

• projektor działał = the projector worked
• drukarka działała = the printer worked
• urządzenie działało = the device worked
• projektory działały = the projectors worked

Also note that the verb agrees with projektor, not with the speaker. Even if the speaker is female, it is still projektor nie działał, because projektor is masculine.

Od razu is a very common fixed phrase meaning:

• right away
• immediately
• straight away
• in negative contexts, often at first or not immediately

So:

• projektor nie działał od razu

means the projector didn’t work immediately or wasn’t working at first.

Examples:

• Zrobię to od razu. = I’ll do it right away.
• Nie zrozumiałem od razu. = I didn’t understand immediately.
• Komputer nie działał od razu. = The computer didn’t work right away / at first.

Even though it literally looks like from once/immediately, it should be learned as a whole expression.

This is about aspect, which is very important in Polish.

• działać = imperfective = to work, to function, to be working
• zadziałać = perfective = to start working, to work successfully, to take effect

In this sentence:

• projektor nie działał od razu

the speaker is describing a state: at first, the projector was not functioning.

If you said:

• projektor nie zadziałał od razu

that would focus more on the moment of starting or kicking in: it didn’t start working immediately.

Both can be possible in real life, but they are not identical:

• nie działał od razu = it wasn’t working at first
• nie zadziałał od razu = it didn’t start working right away

Ale means but, and it introduces a clear contrast:

• I have twenty-one slides,
• but the projector did not work right away.

That contrast is why ale is very natural here.

The comma is standard in Polish before ale when it joins two clauses:

• Na prezentacji mam dwadzieścia jeden slajdów, ale projektor nie działał od razu.

You might also see a in other sentences, but a is often softer and can mean and / while / whereas, not always a strong but.

So here:

• ale = stronger contrast
• comma before ale = correct standard punctuation

Yes. Polish word order is flexible, because endings carry a lot of the grammatical meaning.

This sentence can be rearranged, but the emphasis changes.

For example:

• Na prezentacji mam dwadzieścia jeden slajdów...
  Focuses first on the presentation context.

• Mam dwadzieścia jeden slajdów na prezentacji...
  Also understandable, but can sound a bit more like at the presentation event than in the presentation itself.

• W prezentacji mam dwadzieścia jeden slajdów...
  More clearly emphasizes that the slides are inside the presentation file/deck.

Polish often puts known/background information earlier and newer or more important information later, so the original order sounds very natural.