Here, で marks the place or setting where the action happens.
So in パーティーで乾杯した, で means something like:
It tells you where the toast happened.
Japanese often uses で for the place where an action is carried out.
Compare:
に is often used for:
Since 乾杯した is an action happening in a location, で is the natural choice.
乾杯した comes from:
So:
That means the sentence is in the plain past tense.
Japanese often leaves out the subject when it is clear from context.
So パーティーで乾杯した。 could mean:
The exact subject depends on the situation. This is completely normal in Japanese.
Yes. In Japanese, this is a perfectly normal complete sentence.
Japanese frequently omits:
As long as the listener can infer the missing information, the sentence sounds natural.
With many suru-verbs, the を can be omitted.
So both of these are possible:
The version without を is very common and natural.
In everyday Japanese, 乾杯した is simpler and more typical.
Yes. 乾杯！ is a very common expression meaning Cheers!
So there are two related uses:
In your sentence, it is the verb use: made a toast.
Not necessarily. 乾杯する means to make a toast or say cheers, not specifically to drink alcohol.
Usually people do raise a glass and drink, but the core meaning is the ceremonial toast. The drink could be alcoholic or non-alcoholic depending on the situation.
Yes. Normal Japanese writing would usually be:
パーティーで乾杯した。
The spaces in learning materials are often added just to make the sentence easier to read for beginners.
It is natural and grammatical, but a little short because Japanese often drops context.
Depending on the situation, a speaker might also say:
But パーティーで乾杯した。 by itself is perfectly fine.
Yes. You can say what the toast is for with に.
For example:
In your sentence, that target is simply not mentioned. It just says that the toast happened at the party.