Puella rogat utrum gruem an ciconiam pulchriorem putemus.

It introduces an indirect question with two alternatives: whether ... or.

So:

• utrum gruem an ciconiam = whether the crane or the stork

In natural English, we might often say which instead of whether:

• The girl asks which we think is prettier, the crane or the stork.

But the Latin structure is literally whether A or B.

Because it is part of an indirect question after rogat.

Latin normally uses the subjunctive in indirect questions:

• rogat = she asks
• utrum ... putemus = whether we think

So putemus is not expressing doubt by itself; it is subjunctive mainly because of the construction.

It is built into the verb ending.

• putemus = we may think / we think in this construction
• the ending -mus tells you the subject is we

Latin often leaves subject pronouns unstated because the verb ending already shows the person and number.

There are two different verbs with two different subjects:

• Puella rogat = the girl asks
• putemus = we think

So the girl is the one asking, but we are the ones whose opinion is being asked about.

Because they are the things being judged by putemus.

With putare, Latin can use an object + predicate adjective construction:

• gruem pulchriorem putemus = we think the crane prettier
• ciconiam pulchriorem putemus = we think the stork prettier

So:

• gruem = accusative singular of grus
• ciconiam = accusative singular of ciconia

They are the direct objects of the idea we think/judge.

Pulchriorem is the comparative form of pulcher:

• pulcher = beautiful
• pulchrior = more beautiful / prettier
• pulchriorem = accusative singular masculine/feminine

Here it agrees in case, number, and gender with whichever bird is being considered:

• gruem ... pulchriorem
• ciconiam ... pulchriorem

It means prettier or the prettier one.

Because each alternative is a single bird.

The question is not Are the crane and the stork prettier?
It is Which one do we think is prettier: the crane or the stork?

So the adjective stays singular, agreeing with one bird at a time.

Because the comparison is already clear from utrum ... an.

Latin does not always need quam when two alternatives are directly set against each other. The sentence already presents the two items being compared:

• the crane
• or the stork

So pulchriorem can simply mean the prettier one / more beautiful, with the comparison understood.

You can think of one as understood, yes.

Latin often allows putare to take an object plus a predicate adjective without stating esse:

• gruem pulchriorem putemus = we think the crane prettier
• literally, this is like we think the crane to be prettier

So an English learner may mentally expand it to something like gruem pulchriorem esse putemus, even though the sentence as written does not need esse.

Because it comes from the 3rd-declension noun grus, gruis.

Its accusative singular is:

• gruem

By contrast, ciconia is a 1st-declension noun, so its accusative singular is the more familiar:

• ciconiam

So the two bird names belong to different declensions:

• grus, gruis → gruem
• ciconia, ciconiae → ciconiam

The sentence is arranged as:

• Puella rogat = main clause
• utrum gruem an ciconiam pulchriorem putemus = indirect question

Latin word order is flexible, but this order is very natural. The two alternatives are placed early:

• gruem
• an ciconiam

Then comes the comparative idea:

• pulchriorem

And the verb of thinking comes at the end:

• putemus

That final-verb position is very common in Latin subordinate clauses.

A very natural way is:

• whether we think the crane or the stork prettier
• or more idiomatically, which we think is prettier, the crane or the stork

So even though Latin uses utrum ... an, the smoothest English often uses which when two choices are being compared.