Learn Hardcore German: In der Probezeit arbeite ich aufmerksam und frage oft nach. - During the probation period I work attentively and often ask questions.

Because in takes the dative case when it expresses location or time. Probezeit is feminine, so the dative singular article is der: in der Probezeit.

in die (accusative) would indicate movement into something and isn’t used for a static time frame here.
im = in dem, and dem is the dative for masculine/neuter nouns, so im Probezeit is impossible with a feminine noun.

What exactly is Probezeit?

It’s the “probationary period,” typically at the start of a job, apprenticeship, or similar arrangement. It’s a noun (feminine: die Probezeit) and is written as one word. In contracts you’ll also see time-related variants like während der Probezeit (“during the probationary period”) or legalese innerhalb der Probezeit.

Why does the verb come before ich in “arbeite ich”?

German main clauses are verb-second (V2). When a non-subject element like In der Probezeit is put first, the finite verb still has to be in second position, so the subject ich moves after the verb: In der Probezeit arbeite ich …
Without fronting, you could simply say: Ich arbeite in der Probezeit …

Do I need a comma after “In der Probezeit”?

No. A comma after a short fronted adverbial is normally not used in German. You might add one after a very long or complex introductory phrase for clarity, but here it’s not needed.

What part of speech is aufmerksam here, and why doesn’t it have an ending?

It’s an adjective used adverbially (“attentively”). Adjectives used adverbially in German are uninflected: Ich arbeite aufmerksam.
Compare:

Attributive (with endings): ein aufmerksamer Mitarbeiter
Predicative/adverbial (no endings): Er ist aufmerksam. / Er arbeitet aufmerksam.

Does aufmerksam arbeiten mean “carefully” or “focused”? Are there alternatives?

aufmerksam emphasizes being attentive (noticing things, paying attention). Close alternatives:

sorgfältig = carefully, meticulously
konzentriert = with focus/concentration
gewissenhaft = conscientious(ly)
Pick what best matches the nuance you want.

Is frage … nach one verb? What’s the difference between nachfragen and fragen (nach …)?

Two related patterns exist:

nachfragen (separable verb): ich frage nach = I inquire/follow up (often without naming the topic).
fragen nach + Dativ: ich frage nach dem Plan = I ask about the plan.
Also useful patterns:
Ask a person directly: jemanden (Akk.) fragen (Ich frage meinen Chef.)
Inquire with someone: bei + Dativ (Ich frage bei der Personalabteilung nach.)

Why is there no object after nach in “frage oft nach”?

Because with the separable verb nachfragen, you can naturally omit the object when it’s understood: Ich frage oft nach ≈ “I often follow up/ask for clarification.”
With fragen nach, you usually name the thing: Ich frage oft nach dem Status. In casual speech, the object can be omitted if context makes it clear.

Where does oft go in “frage oft nach”? Could I say “frage nach oft” or move it elsewhere?

In main clauses with a separable verb, the particle goes to the end and adverbs like oft sit before it: Ich frage oft nach.
Correct variants for emphasis:

Oft frage ich nach. (fronted adverb, with inversion)
Ich frage oft nach. (neutral)
Do not say Ich frage nach oft.

How would I say this in the past?

Perfekt (common in speech): Ich habe aufmerksam gearbeitet und oft nachgefragt.
Präteritum (common in writing): Ich arbeitete aufmerksam und fragte oft nach.
If you’re using fragen nach + Dativ: Ich habe oft nach dem Status gefragt.

What happens in a subordinate clause?

Subordinators like weil/dass send the finite verb to the end. With a separable verb, the particle reattaches:

…, weil ich in der Probezeit aufmerksam arbeite und oft nachfrage.
If you name the object: …, weil ich oft nach dem Status frage.

Is it okay to write “… und ich frage oft nach” instead of “… und frage oft nach”?

Yes. Both are correct. German often omits the repeated subject in coordinated predicates: … arbeite ich … und frage ….
You can include it for emphasis or clarity: … und ich frage … (slightly more formal/explicit).

Could I say “Ich arbeite aufmerksam in der Probezeit” instead? What’s the preferred order?

Yes, that’s grammatical. The common neutral order for adverbials is Time–Manner–Place. So many speakers prefer: Ich arbeite in der Probezeit aufmerksam. Fronting the time phrase (In der Probezeit …) simply gives it emphasis.

Can I use während instead of in?

Yes: Während der Probezeit arbeite ich aufmerksam ….
Nuance: während explicitly marks “during” and can sound a bit more formal. In der Probezeit is simple and idiomatic. Legal texts may prefer innerhalb der Probezeit.

Why is there no comma before und?

Here und links two predicates sharing the same subject inside one main clause, so no comma is used. If you had two full main clauses (each with its own subject and verb), a comma is still usually not used before und in German, though it can be optional in long, complex sentences for readability.

Any capitalization points I should notice?

Probezeit is a noun, so it’s capitalized.
ich is normally lowercase unless it starts the sentence.
Adverbs/adjectives like aufmerksam, oft are lowercase.

How do I mention both the person I ask and the thing I ask about?

Combine the patterns:

Ich frage meinen Chef nach dem Status. (Akk. person + nach + Dat. thing)
With nachfragen: Ich frage bei meinem Chef nach dem Status nach. (colloquial but common; often you’ll drop one of the two if context is clear)

Is there a nuance difference between oft, häufig, öfter, and öfters?

oft and häufig ≈ often/frequently (both standard; häufig is a bit more formal).
öfter = more often (comparative of oft).
öfters is colloquial/regional; in standard usage prefer öfter.