Našel páku, aniž by použil nápovědu.

Because páku is the accusative singular form of páka.

The verb najít / našel (to find / found) usually takes a direct object in the accusative case, so:

• páka = lever, as the dictionary form
• našel páku = found a/the lever

This is very common in Czech:

• viděl ženu = he saw a woman
• koupil knihu = he bought a book
• našel páku = he found a lever

Since páka is a feminine noun, its accusative singular ends in -u.

For the same basic reason: nápovědu is also accusative singular.

The verb použít (to use) takes a direct object in the accusative:

• nápověda = hint, clue, help prompt
• použil nápovědu = he used a hint / he used the help

So in the clause:

• aniž by použil nápovědu

  the noun nápovědu is the thing being used, so it must be in the accusative case.

Aniž by means without doing something or without the fact that someone did something.

So:

• Našel páku, aniž by použil nápovědu. = He found the lever without using the hint.

A very useful way to understand it is:

• aniž by + verb = without + -ing

Examples:

• Odešel, aniž by něco řekl. = He left without saying anything.
• Udělala to, aniž by požádala o pomoc. = She did it without asking for help.

It is a very common and useful structure in Czech.

Here by is part of the fixed construction aniž by.

It is related to the conditional particle by, but in this pattern you should mostly learn it as a unit:

• aniž by udělal...
• aniž by řekl...
• aniž by použil...

In English, you usually do not translate by separately. The whole phrase simply means:

• without doing
• without having done

So don’t try to translate it word-for-word as without would use or anything like that. In this sentence, aniž by použil nápovědu naturally means without using the hint.

Yes, použil is the -l form of použít, which is the form also used in the past tense. But after aniž by, Czech uses this kind of form in a special subordinate construction.

So although it looks like a past-tense form, in this sentence it is best understood as part of the pattern:

• aniž by + l-form

Examples:

• aniž by věděl = without knowing
• aniž by to zkusil = without trying it
• aniž by použil nápovědu = without using the hint

So the key point is:

• don’t interpret použil here as a separate simple past like he used
• interpret aniž by použil together as without using

Czech often leaves out subject pronouns when they are clear from the verb form.

Here:

• našel = found, masculine singular past
• použil = used, masculine singular past/l-form

That tells us the subject is understood as he (or sometimes a masculine noun mentioned earlier).

So Czech does not need to say on (he) here. A full version with the pronoun would be possible, but usually unnecessary:

• (On) našel páku, aniž by použil nápovědu.

In natural Czech, omitting on is normal unless you want emphasis or contrast.

Because they agree with the understood subject.

In the past tense and in these -l forms, Czech shows gender and number:

• našel / použil = masculine singular
• našla / použila = feminine singular
• našlo / použilo = neuter singular
• našli / použili = plural masculine animate
• našly / použily = plural non-masculine-animate

So this sentence specifically suggests a masculine singular subject:

• Našel páku, aniž by použil nápovědu. = He found the lever without using the hint.

If the subject were female, it would be:

• Našla páku, aniž by použila nápovědu.

Because našel comes from the perfective verb najít, which focuses on a completed result: he successfully found it.

Compare:

• hledal páku = he was looking for the lever / he searched for the lever
• našel páku = he found the lever

So našel tells us the action reached its goal.

The same applies to použil from použít, which is also perfective. In this sentence, the meaning is about whether he used the hint at all as a completed action:

• he found the lever
• and he did so without using the hint

Yes, normally yes.

Aniž by introduces a subordinate clause, and in Czech subordinate clauses are usually separated by a comma.

So:

• Našel páku, aniž by použil nápovědu.

That comma is standard and expected.

Yes. A common alternative is to use a noun phrase instead of aniž by:

• Našel páku bez použití nápovědy.

This also means He found the lever without using the hint.

Compare the two:

• Našel páku, aniž by použil nápovědu.
  More clause-based, very natural.
• Našel páku bez použití nápovědy.
  Slightly more compact, a bit more noun-based or formal in style.

Both are correct.

It depends on context.

Nápověda can mean:

• hint
• clue
• prompt
• sometimes help in a game, puzzle, app, or interface

So in this sentence, if the context is a puzzle, game, or task, nápovědu is often best translated as:

• a hint
• the hint
• help

For example:

• in a game: without using a hint
• in software: without using the help prompt
• in a puzzle: without using a clue

So the Czech grammar stays the same, but the best English word depends on the situation.