Im dłuższa jest trasa, tym bardziej jesteśmy zmęczeni.

This is a very common Polish pattern used to express the more..., the more... or the longer..., the more....

So:

• im dłuższa jest trasa = the longer the route is
• tym bardziej jesteśmy zmęczeni = the more tired we are

You can think of it as a paired structure:

• im introduces the first comparison
• tym introduces the result or parallel comparison

A good basic model is:

• Im X, tym Y. = The more/greater X, the more/greater Y.

Examples:

• Im więcej pracuję, tym mniej mam czasu. = The more I work, the less time I have.
• Im szybciej jedziemy, tym wcześniej dojedziemy. = The faster we go, the earlier we’ll arrive.

Because dłuższa agrees with trasa.

The noun trasa is:

• feminine
• singular

So the adjective must also be:

• feminine
• singular

That gives:

• długi = long
• dłuższy = longer

Agreement forms:

• dłuższy = masculine singular
• dłuższa = feminine singular
• dłuższe = neuter singular or non-masculine-personal plural

Since trasa is feminine, you need dłuższa.

Both are understandable, but the version in your sentence sounds natural in this im ..., tym ... structure.

Polish word order is much freer than English, and in comparative patterns like this, speakers often place the descriptive word earlier:

• Im dłuższa jest trasa, tym bardziej jesteśmy zmęczeni.

This highlights dłuższa right away, which fits the meaning pattern well.

You may also hear:

• Im trasa jest dłuższa, tym bardziej jesteśmy zmęczeni.

This is also correct, but it may sound a bit more neutral or less elegant depending on context.

So the original wording is not random—it is a natural stylistic choice.

Yes, very often it can.

You can say:

• Im dłuższa trasa, tym bardziej jesteśmy zmęczeni.

This is very natural and probably even a bit more compact in everyday Polish.

Why is that possible? Because Polish often leaves out to be in structures where the meaning is still obvious.

So both are correct:

• Im dłuższa jest trasa, tym bardziej jesteśmy zmęczeni.
• Im dłuższa trasa, tym bardziej jesteśmy zmęczeni.

The version with jest is slightly more explicit.

Because bardziej means more, and here it modifies the adjective zmęczeni (tired).

So:

• zmęczeni = tired
• bardziej zmęczeni = more tired

This is the normal way to form the comparative with many adjectives in Polish, especially when there is no commonly used simple comparative form in that context.

It works like this:

• bardziej szczęśliwy = happier / more happy
• bardziej interesujący = more interesting
• bardziej zmęczeni = more tired

In English, more tired is also perfectly normal, so this lines up quite well.

Not in this sentence if you want the meaning the more tired we are.

You need a comparative idea in the second half, so bardziej is important.

• tym bardziej jesteśmy zmęczeni = the more tired we are

If you removed bardziej, the sentence would lose the parallel comparison and sound incomplete or unnatural in this pattern.

The structure im ..., tym ... usually expects some kind of matching comparison:

• im dłuższa... tym bardziej...
• im więcej... tym mniej...
• im szybciej... tym lepiej...

Because zmęczeni must agree with my (we) in gender/number.

In Polish plural, past-tense forms and predicate adjectives often distinguish:

• masculine personal: zmęczeni
• non-masculine-personal: zmęczone

So:

• jesteśmy zmęczeni is used for a group containing at least one male, or for a mixed group
• jesteśmy zmęczone is used for a group of females only

Because the sentence says jesteśmy zmęczeni, it suggests the speakers are:

• all male, or
• a mixed group

If the group were only women, you would say:

• Im dłuższa jest trasa, tym bardziej jesteśmy zmęczone.

Historically it is related, yes, but in this sentence it functions as part of the fixed comparative pattern im ..., tym ....

So for a learner, the most useful thing is to treat it as a set expression:

• im ..., tym ... = the ..., the ...

You do not need to analyze tym here as a separate ordinary demonstrative every time. In this structure, it acts more like a grammatical marker.

That said, you will also see tym elsewhere in other meanings, for example:

• tym razem = this time
• z tym człowiekiem = with this man/person

So it is the same form, but in your sentence it belongs to a special comparison pattern.

Yes, normally yes.

The sentence has two linked parts:

• Im dłuższa jest trasa
• tym bardziej jesteśmy zmęczeni

These are separated by a comma in standard writing:

• Im dłuższa jest trasa, tym bardziej jesteśmy zmęczeni.

This is the normal punctuation for this correlative structure.

Yes. Very often.

You can use it with:

• adjectives
• adverbs
• quantities
• whole clauses

Examples:

• Im więcej czytasz, tym więcej rozumiesz.
  The more you read, the more you understand.

• Im szybciej wyjedziemy, tym lepiej.
  The sooner we leave, the better.

• Im mniej mówisz, tym lepiej.
  The less you say, the better.

• Im trudniejsze zadanie, tym większa satysfakcja.
  The harder the task, the greater the satisfaction.

So the pattern is extremely useful and worth memorizing as a whole.

People would absolutely say it. It is natural Polish.

It sounds:

• correct
• standard
• normal in both speech and writing

The slightly shorter version without jest may sound a bit more conversational:

• Im dłuższa trasa, tym bardziej jesteśmy zmęczeni.

But the version you were given is fully natural too.

No. In this structure, im does not mean if or when.

That is an easy mistake for English speakers, because the sentence can feel a bit conditional. But grammatically, this is not really if X, then Y.

It is specifically a comparative correlation:

• im ... tym ... = the more..., the more...

So it is better to learn it as one fixed pattern rather than trying to match im to a single English word by itself.