Mi proponas, ke ni trinku teon en la kafejo post la laboro.

Ke introduces a subordinate clause, much like English that.

So:

Mi proponas = I propose
ke ni trinku teon... = that we drink tea... / that we should drink tea...

In Esperanto, if you follow a verb like proponi with a full clause that has its own subject and verb, ke is the normal way to connect it.

Trinku is the -u form, often called the volitive mood. It is used for commands, wishes, requests, suggestions, and proposals.

Here, the speaker is not stating a fact. The speaker is proposing an action. That is why Esperanto uses ke ni trinku.

Compare:

• ni trinkas = we are drinking / we drink
• ni trinkos = we will drink
• ni trinku = let us drink / that we drink / that we should drink

Even though the action will happen later, Esperanto uses -u because the important idea is proposal, not simple future time.

For an English speaker, that is a useful way to think about it.

Esperanto does not have a separate traditional subjunctive system like some European languages, but the -u form often covers meanings that English expresses with things like:

• let us...
• should...
• that we... in a more formal style

So ke ni trinku is very close in function to a subjunctive-style idea, even though Esperanto usually calls it the volitive form.

No. The ending -u does not show the subject by itself.

So trinku alone just means something like drink! / let ... drink, depending on context. You normally need the subject to make it clear:

• mi trinku = let me drink
• li trinku = let him drink
• ni trinku = let us drink

That is why ni is important here.

Because teon is the direct object.

The tea is what is being drunk, so it takes the accusative ending -n:

• teo = tea
• teon = tea as the direct object

Compare:

• Ni trinkas teon = We drink tea
• Ni trinkas kafon = We drink coffee

This -n is one of the most basic and important Esperanto endings.

Because en shows location, while al shows movement toward something.

Here, the sentence is about the place where the tea-drinking happens:

• en la kafejo = in / at the café

If the sentence were about going there, then al would be appropriate:

• Ni iru al la kafejo = Let’s go to the café

So:

• trinki en la kafejo = drink in/at the café
• iri al la kafejo = go to the café

La means the, but Esperanto does not always use articles exactly the same way English does.

In la kafejo, it can mean:

• a specific café already known in context, or
• the relevant café in the situation

In post la laboro, Esperanto very naturally uses la, even where English often just says after work with no article. It has the sense of after the work / after the workday / after finishing work.

So this is a good example of article usage not matching English one-for-one.

Yes, that is also possible.

That version uses the infinitive trinki and means something like:

I propose drinking tea in the café after work.

The version with ke ni trinku is a little more explicit, because it clearly says who is supposed to drink the tea: we.

So the difference is roughly:

• Mi proponas trinki... = I propose drinking...
• Mi proponas, ke ni trinku... = I propose that we drink...

Both are grammatical.

Ni trinku... is the direct suggestion:

Let’s drink tea...

Mi proponas, ke ni trinku... is more explicit and a bit more indirect:

I propose that we drink tea...

So the first one simply makes the suggestion. The second one talks about the act of making a proposal.

Not completely. Esperanto word order is fairly flexible, because endings and function words show the grammar.

The given sentence has a very natural neutral order:

Mi proponas, ke ni trinku teon en la kafejo post la laboro.

But you could move parts around for emphasis, for example:

Post la laboro, mi proponas, ke ni trinku teon en la kafejo.

That puts more focus on after work. The sentence still works because the grammar is clear.