Byl liják, přesto jsme přišli.

Because liják is a masculine inanimate noun, so the past tense of být has to agree with it:

• byl = masculine singular
• byla = feminine singular
• bylo = neuter singular

So:

• Byl liják = literally There was a downpour

English uses there was, but Czech often just uses byl/byla/bylo + noun.

Liják means a downpour, heavy rain, or cloudburst. It is stronger and more vivid than just déšť (rain).

So:

• déšť = rain
• liják = a really heavy rain / downpour

It is a masculine inanimate noun.

Czech usually does not need a dummy subject like English there.

English says:

• There was a downpour.

Czech simply says:

• Byl liják.

That is completely natural. Czech often expresses existence or occurrence without any equivalent of English there.

Přesto means nevertheless, all the same, even so, or despite that.

In this sentence, it introduces a contrast:

• Byl liják, přesto jsme přišli.
• There was a downpour; nevertheless, we came.

It connects the first idea and the unexpected second result.

Because přesto expresses a stronger idea of despite that fact.

Compare:

• ale = but
• přesto = nevertheless / even so / despite that

So přesto fits well when the second clause happened in spite of the first one.

Very roughly:

• Byl liják, ale přišli jsme. = There was a downpour, but we came.
• Byl liják, přesto jsme přišli. = There was a downpour; nevertheless, we came.

The second version sounds a bit more explicit and emphatic.

Because Czech often omits subject pronouns when they are not needed.

The verb already tells you the person:

• jsme přišli = we came

So my (we) is unnecessary unless you want emphasis or contrast.

Compare:

• Jsme přišli. = We came.
• My jsme přišli. = We came. / As for us, we came.

In your sentence, the neutral version without my is the most natural.

Because jsme is a clitic, and Czech clitics usually go in the second position of the clause.

In the clause:

• přesto jsme přišli

přesto takes the first position, so jsme naturally comes right after it.

That is why this order is standard and natural.

• Přesto jsme přišli. = natural
• Přesto přišli jsme. = unnatural in normal speech

Czech word order is flexible, but clitics follow special placement rules.

Přišli is the past active participle used to form the past tense with jsme.

It tells you two things here:

1. past tense
2. plural masculine personal / mixed-gender group

So:

• jsme přišli = we came

If the group were entirely female, it would be:

• Byl liják, přesto jsme přišly.

So the ending -i in přišli usually shows a group containing at least one male, or a mixed group.

Because přišli comes from přijít, which is perfective. It expresses a completed event: we arrived / we came.

That fits the meaning here: despite the downpour, the action was completed.

Compare:

• přišli = we came / we arrived once, successfully
• přicházeli = we were coming / used to come / were in the process of coming
• chodili = we used to go / used to come regularly

So přišli is the natural choice for a single finished action.

Yes, the comma is natural and standard here.

You have two finite clauses:

• Byl liják
• přesto jsme přišli

They are joined in a contrastive relationship, so the comma is used.

In English you might also use a semicolon or a conjunction, but in Czech the comma is standard in this kind of sentence.

Yes. Several alternatives are possible, depending on style and emphasis. For example:

• Byl liják, ale přišli jsme. = There was a downpour, but we came.
• I přes liják jsme přišli. = We came despite the downpour.
• Přestože byl liják, přišli jsme. = Although there was a downpour, we came.
• Pršelo, přesto jsme přišli. = It was raining, nevertheless we came.

Your original sentence is correct and natural; it just has a slightly more formal or written feel because of přesto.