Postranní čára vede k zadní čáře.

Because it comes after the preposition k, and k requires the dative case.

So:

• čára = nominative singular
• k čáře = dative singular

That is why the sentence has k zadní čáře, not k zadní čára.

A short declension of čára may help:

• nominative: čára
• genitive: čáry
• dative: čáře
• accusative: čáru
• locative: čáře
• instrumental: čárou

K takes the dative.

Its basic meaning is to / toward / up to. In this sentence, it marks the destination or endpoint:

• k zadní čáře = to the back line

This is a very common pattern in Czech, so it is worth memorizing:

• k + dative

In Czech, just like in English, a road, path, line, or direction can figuratively lead somewhere.

So vede here does not mean that the line is actively guiding a person. It means something like:

• goes to
• runs to
• extends to
• leads to

With diagrams, maps, sports courts, or geometric descriptions, this is very natural Czech.

Vede is the 3rd person singular present form of vést.

So:

• infinitive: vést
• vede = it leads / it goes / it runs

Since the subject is postranní čára and that is singular, the verb is singular too:

• čára vede
• not čára vedou

Yes, the adjective does agree with the noun in case, number, and gender. The reason it still looks like zadní is that adjectives ending in -í often have the same form in several cases.

Here:

• zadní čára = nominative singular feminine
• k zadní čáře = dative singular feminine

The noun changes clearly from čára to čáře, but the adjective zadní happens to look the same.

The same is true of postranní: it is also an -í adjective.

They are adjectives.

They describe the noun čára:

• postranní čára = side line / sideline
• zadní čára = back line / baseline

Both adjectives agree with čára, which is a feminine singular noun.

Also, both are so-called soft adjectives ending in -í, which is why their endings may seem less variable than adjectives like dobrý.

Because vést k is the natural pattern here.

Very roughly:

• k = to / toward
• do = into
• na = onto / to a surface / to certain destinations

Here the idea is not entering something or getting onto a surface. It is simply that one line reaches or connects to another line. So vede k is the idiomatic choice.

Usually vede is better here.

• jít means to go and is mainly used for people, animals, or things moving.
• vést is commonly used for roads, paths, lines, corridors, and routes that lead somewhere.

So in a sentence like this, vede sounds natural, while jde would usually sound odd.

No, Czech word order is fairly flexible, but the original sentence is the most neutral and straightforward:

• Postranní čára vede k zadní čáře.

You could change the order for emphasis, for example:

• K zadní čáře vede postranní čára.

That puts more focus on where it leads.

Still, for a learner, the original order is the best default: subject + verb + destination

A useful learner-friendly approximation is:

PO-stran-nee CHAA-ra VE-de k ZAD-nyee CHAA-rzhe

A few key points:

• stress is usually on the first syllable of each Czech word
• č sounds like ch in chair
• á is a long vowel: aa
• í is a long ee
• ř is the famous Czech sound with no exact English equivalent; an approximate learner version is something like rzh

So:

• čára ≈ CHAA-ra
• čáře ≈ CHAA-rzhe

Very likely, yes, because postranní čára and zadní čára are common terms for court or field markings.

But grammatically, the sentence could also be used in any description of lines in a diagram, plan, or layout. The structure is not limited to sports. The sports context just makes the vocabulary especially natural.