Learn Hardcore German: Bitte berücksichtigen Sie die Frist. - Please take the deadline into account.

Sie is the formal you in German and is always capitalized in standard writing to distinguish it from sie (she/they). It can refer to one person or more than one; the form is the same. Only context tells you whether it's one or many.

How is the imperative formed with Sie in Bitte berücksichtigen Sie ...?

For the polite imperative, German uses the verb in its infinitive form + Sie: Berücksichtigen Sie ...; Gehen Sie ...; Nehmen Sie ....
Bitte is a politeness marker and can be added; it does not change the imperative formation.
A more indirect, very polite alternative is: Würden Sie bitte die Frist berücksichtigen?

What case is die Frist, and why is it die?

Frist is a feminine noun (gender: feminine).
It is the direct object of berücksichtigen, so it stands in the accusative case.
Feminine nouns have die for both nominative and accusative singular, hence die Frist is correct (not der/dem Frist).

Is berücksichtigen the same as beachten or einhalten?

Not exactly; they differ in nuance:

berücksichtigen = to take into account/consider as a factor. You might still decide against it, but you have factored it in.
beachten = to heed/observe. Stronger than merely considering; it implies following a rule or instruction.
einhalten (eine Frist) = to meet/comply with a deadline. So depending on intent, you might also see: Bitte beachten Sie die Frist or Bitte halten Sie die Frist ein.

Where can I place bitte in this sentence? Does word order change the tone?

Common options (all natural):

Bitte berücksichtigen Sie die Frist.
Berücksichtigen Sie bitte die Frist.
Berücksichtigen Sie die Frist, bitte. Differences are minimal; sentence-initial Bitte sounds slightly more formal or businesslike.

Is berücksichtigen separable? How do I conjugate it in the past?

It is inseparable (prefix be- is inseparable).
Past participle: berücksichtigt (no ge- is added).
Perfekt: Ich habe die Frist berücksichtigt.
Präteritum (simple past): Ich berücksichtigte die Frist. (rarer in everyday speech)

What exactly does Frist mean? How is it different from Termin or Datum?

Frist = a deadline or time limit by which something must be done.
Termin = an appointment/meeting or a scheduled date/time.
Datum = a calendar date. Example: You have a Termin on May 5; the Frist to submit documents is May 1; that date (May 1) is the Datum of the deadline.

Can I drop Sie or the article and say Bitte Frist berücksichtigen?

You will see this ellipsis on signs, forms, or notes, but it is telegraphic style. In full, polite sentences—especially in emails or letters—use the normal imperative: Bitte berücksichtigen Sie die Frist.

Is a comma after Bitte required?

No. In Bitte berücksichtigen Sie die Frist you normally do not use a comma. You may insert one if bitte functions as a clear interjection (e.g., Bitte, nehmen Sie Platz.), but it is not needed here.

How would I say this informally to one person or to a group?

To one person (du): Berücksichtige die Frist, bitte.
To a group you know (ihr): Berücksichtigt die Frist, bitte. The polite/business form remains Berücksichtigen Sie die Frist, bitte.

What are common collocations with Frist?

eine Frist setzen/verlängern/verkürzen (set/extend/shorten a deadline)
eine Frist beachten/einhalten (heed/meet a deadline)
eine Frist versäumen/verpassen (miss a deadline)
Fristablauf (expiry of a deadline)

How do I pronounce the sentence?

Approximate IPA: [ˈbɪtə bəˈʁʏkzɪçtɪɡən ziː diː fʁɪst] Tips:

Stress on rück in berücksichtigen.
ch here is the soft ich-sound [ç].
Sie is [ziː]; Frist has a short, crisp [ɪ].

Why not Berücksichtigen Sie bitte der Frist?

Because berücksichtigen takes a direct object in the accusative. der Frist is dative; the correct form is accusative feminine singular: die Frist.

Is there a more formal noun-based alternative?

Yes, in very formal writing you might see: Bitte um Berücksichtigung der Frist. That’s a nominalized request meaning roughly “Request for consideration of the deadline.”

What’s the difference between Rücksicht nehmen auf and berücksichtigen?

They are closely related in meaning.

Rücksicht auf etwas nehmen = to take consideration of something (more idiomatic in some contexts, slightly more formal/set phrase).
etwas berücksichtigen = to take something into account (straightforward verb). In most contexts with deadlines, you’ll prefer beachten/einhalten or simply keep berücksichtigen as in the original sentence.