A kenu anélkül sodródott az áramlattal, hogy eveztünk volna.

It is the standard Hungarian pattern for without doing something or without someone doing something when a whole clause follows.

• nélkül = without
• anélkül = without that / without it
• hogy = that

So anélkül ..., hogy eveztünk volna is literally something like without it being the case that we had rowed, but in natural English it corresponds to without rowing or without us rowing.

Hungarian often uses this full clause structure where English would prefer a gerund.

After anélkül, hogy, Hungarian normally uses a finite clause, not an infinitive.

So instead of an English-style structure like without to row or even without rowing, Hungarian says:

• anélkül, hogy eveztünk volna

The form eveztünk volna is a conditional form that fits this construction and signals that the rowing did not happen.

Using plain eveztünk would sound wrong here, because it would state the rowing as a normal past fact, while this sentence needs the idea without having rowed / without our rowing.

Using evezni would also not fit after hogy, because hogy introduces a full clause, and a full clause needs a finite verb.

Volna is the marker of the conditional.

In eveztünk volna, it helps create what is often called the past conditional:

• eveztünk = we rowed
• eveztünk volna = we would have rowed

In this sentence, though, it is not mainly giving an English-style would have meaning by itself. It is part of the normal grammar after anélkül, hogy, where Hungarian uses the conditional to express an action that did not take place.

So the point is not really we would have rowed, but rather without our having rowed.

Because the sentence contains two different clauses, and each clause can have its own subject.

Main clause:

• A kenu sodródott = the canoe drifted

Subordinate clause:

• hogy eveztünk volna = that we would have rowed / that we had rowed

So the canoe is the subject of sodródott, but the hidden subject of eveztünk volna is we.

Hungarian often leaves subject pronouns out when the verb ending already shows the person.
Here -tünk clearly means we, so mi is unnecessary.

If you wanted to emphasize it, you could say:

• ... hogy mi eveztünk volna

Sodródott comes from sodródik, which means to drift, to be carried along, or to be swept along.

It suggests movement that is not actively controlled by the subject. In this sentence, the canoe is being carried by the water rather than being propelled by rowing.

So sodródott is a very good choice here because it contrasts nicely with eveztünk volna:

• sodródott = it drifted
• eveztünk volna = we would have rowed / had rowed

The whole sentence therefore emphasizes that the canoe moved passively, not through our effort.

Áramlat means current.
The ending -val / -vel usually means with.

So:

• áramlat + -val = áramlattal

The reason it becomes -tal instead of -val is sound assimilation. In Hungarian, -val/-vel changes to match the final consonant of the word before it.

Because áramlat ends in t, the v disappears and the t doubles:

• áramlat + val → áramlattal

Here az áramlattal means with the current or along with the current.

Because Hungarian usually uses the definite article where English also often would:

• az áramlattal = with the current

The form is az rather than a because the next word begins with a vowel:

• a kenu
• az áramlat

This is just the normal a / az alternation in Hungarian.

Because hogy introduces a subordinate clause, and Hungarian normally separates such clauses with a comma.

So in:

• A kenu anélkül sodródott az áramlattal, hogy eveztünk volna.

the comma marks the start of the hogy clause.

This is standard Hungarian punctuation.

Yes.

• nélkül simply means without
• anélkül is a more specific form meaning something like without that / without it

In modern Hungarian, anélkül is a fixed word, and it is very commonly used in the pattern:

• anélkül, hogy ...

By contrast, nélkül can directly follow a noun:

• evezés nélkül = without rowing
• segítség nélkül = without help

So a useful shortcut is:

• noun + nélkül
• anélkül, hogy + clause

Yes. Hungarian word order is flexible, and changing it usually changes emphasis rather than basic meaning.

This sentence has a fairly neutral structure:

• A kenu = topic
• anélkül = sets up the without idea
• sodródott = verb
• az áramlattal = with the current

You could move parts around, but the emphasis would shift. For example:

• Anélkül sodródott a kenu az áramlattal, hogy eveztünk volna.

This puts a bit more focus on the without idea.

So the original order is natural, but not the only possible one.