Po prysznicu używam suszarki.

Because the preposition po usually requires the locative case when it means after.

• prysznic = the basic dictionary form
• po prysznicu = after the shower

So:

• po obiedzie = after lunch
• po pracy = after work
• po prysznicu = after the shower

This is just a case change required by po.

Because the verb używać normally takes the genitive case in Polish.

So:

• suszarka = dictionary form
• używam suszarki = I use a dryer / hairdryer

This is one of those verb patterns you simply learn:

• używać + genitive
• szukać + genitive
• potrzebować + genitive

A learner might expect suszarkę because English uses a direct object with use, but Polish does it differently with używać.

Literally, suszarka means a dryer. In this sentence, because of the context after the shower, it most naturally means a hairdryer.

If you want to be more explicit, you can say:

• suszarka do włosów = hairdryer

So the sentence could also be:

• Po prysznicu używam suszarki do włosów.

But in everyday speech, suszarka alone is often enough if the meaning is obvious.

It can be understood as either, depending on context. Very often in natural English translation, it is best rendered as:

• After showering, I use a hairdryer.

Literally, though, Polish is saying:

• After the shower, I use a dryer.

Polish often uses a noun phrase like this where English might prefer a verb form such as after showering.

Because Polish verbs usually already show who is doing the action.

• używam = I use

So ja is not necessary. Polish often drops subject pronouns unless they are needed for emphasis, contrast, or clarity.

Compare:

• Używam suszarki. = I use a hairdryer.
• Ja używam suszarki, a on ręcznika. = I use a hairdryer, and he uses a towel.

So in your sentence, leaving out ja is completely normal.

Używam is the 1st person singular present tense form of używać.

So:

• używam = I use
• używasz = you use
• używa = he/she/it uses
• używamy = we use
• używacie = you all use
• używają = they use

In this sentence, it usually describes a regular habit or routine:

• After the shower, I use a hairdryer.

Most naturally, it describes a habit or usual routine.

• Po prysznicu używam suszarki. = After showering, I use a hairdryer.

Without extra context, this sounds like a general statement about what you normally do. Polish present tense can sometimes describe current action too, but here the sentence strongly suggests a routine.

Yes. Polish word order is fairly flexible.

These are all possible:

• Po prysznicu używam suszarki.
• Używam suszarki po prysznicu.
• Suszarki używam po prysznicu.

They all mean roughly the same thing, but the emphasis changes.

• Po prysznicu używam suszarki. → emphasis on after the shower
• Używam suszarki po prysznicu. → more neutral in some contexts
• Suszarki używam po prysznicu. → emphasis on the dryer

The version you were given is very natural.

Yes, it is very natural. Polish often prefers short noun-based expressions with prepositions.

So instead of building something like after taking a shower, Polish commonly says:

• po prysznicu = after the shower / after showering
• po kąpieli = after the bath
• po kolacji = after dinner

This is one of the normal, everyday ways Polish expresses after doing something.

Yes, you can, but the structure changes.

• używam suszarki
• korzystam z suszarki

Both can mean I use a dryer, but:

• używać takes genitive
• korzystać z takes z + genitive

So:

• suszarka → suszarki
• z suszarki = from/of the dryer, as required by korzystać z

In this sentence, używam suszarki is simple and natural.

It can refer to both, depending on context.

• wziąć prysznic = to take a shower
• pod prysznicem = in the shower
• po prysznicu = after the shower / after showering

So in your sentence, po prysznicu is best understood as after having a shower rather than only after the physical shower unit.