Learn Hardcore Indonesian: Kami mengukur jarak lari dengan aplikasi agar data latihan rapi. - We measure the running distance with an app so the training data is neat.

In Kami mengukur jarak lari..., the speaker is talking about their group only, not including the listener.
If the speaker wants to include the listener (e.g. a coach talking to their team), they would more naturally say Kita mengukur jarak lari....

Why is mengukur used instead of just ukur?

Mengukur is the active verb form meaning to measure.
The base word ukur is a root word and is usually not used by itself as a verb in normal sentences; it appears in dictionary form, in compounds, or with prefixes/suffixes.

So Kami mengukur jarak lari... = We measure the running distance...
Using Kami ukur jarak lari... sounds incomplete or overly casual/non‑standard.

What exactly does jarak lari mean? Why not jarak berlari?

Jarak lari is literally running distance:

jarak = distance
lari = run / running

In Indonesian, two nouns (or a noun + a verb used as a noun) are often put together where English might use of or -ing, so jarak lari ≈ distance of (our) running / running distance.

Jarak berlari is grammatically possible but sounds less natural and a bit stiff; speakers usually say jarak lari, jarak tempuh lari, or jarak lari kami instead.

What does dengan aplikasi literally mean? Could I say menggunakan aplikasi instead?

Dengan aplikasi literally means with an app (using an app as a tool).
Yes, you can also say Kami mengukur jarak lari menggunakan aplikasi..., which means We measure the running distance using an app....

Both are natural.

dengan aplikasi is slightly shorter and more casual.
menggunakan aplikasi is a bit more explicit and slightly more formal.

How is agar used here, and what’s the difference between agar, supaya, and untuk?

Agar introduces a purpose or desired result: so that, in order that.
So agar data latihan rapi = so that the training data is neat/organized.

Comparison:

agar and supaya are very close in meaning; supaya is often a bit more casual/conversational.
You could say ...dengan aplikasi supaya data latihan rapi with almost no change in meaning.
untuk = for / to, and is usually followed by a verb or noun, e.g.
Kami memakai aplikasi untuk mengukur jarak lari. (We use an app to measure running distance.)
Here, agar is better because it introduces a result clause, not just a purpose with a verb.

What does data latihan mean exactly? Which word modifies which?

Data latihan is a noun–noun phrase:

data = data
latihan = training / practice

In Indonesian, the head noun comes first and the modifier follows, so data latihan = training data (literally: data (of) training).
If you want to be more specific, you can add a pronoun: data latihan kami = our training data.

What does rapi mean here, and is it only for physical tidiness?

Rapi means neat, tidy, orderly, well‑organized.
It can refer to:

physical things: kamar rapi (a tidy room), pakaian rapi (neat clothes)
abstract things: jadwal rapi (well‑organized schedule), data rapi (clean/organized data)

So data latihan rapi = the training data is tidy / well organized.

Why is it data latihan rapi and not rapi data latihan?

In Indonesian, adjectives usually come after the noun they describe.
So:

data latihan rapi = training data (that is) neat
Saying rapi data latihan is not natural in standard Indonesian.

The pattern is: noun + (noun modifier) + adjective, e.g. buku pelajaran baru (new textbook), mobil merah cantik (beautiful red car).

Does this sentence have a tense? How do we know if it’s present, past, or future?

Indonesian verbs don’t change form for tense.
Kami mengukur jarak lari dengan aplikasi agar data latihan rapi can mean:

We measure (habitually) the running distance…
We measured the running distance…
We will measure the running distance…

The tense is understood from context or from time words like kemarin (yesterday), tadi (earlier), nanti (later), setiap hari (every day), etc.
For example: Setiap hari kami mengukur jarak lari... clearly means a habitual action.

Can the subject kami be omitted? Would Mengukur jarak lari dengan aplikasi agar data latihan rapi be correct?

You can drop kami in casual speech or writing when the subject is already clear from context.
Mengukur jarak lari dengan aplikasi agar data latihan rapi is understandable and not ungrammatical, but it sounds like a sentence fragment or a bullet point, e.g. in instructions or a list.

For a full, normal sentence, especially in isolation, Kami mengukur... is more natural and complete.