Learn Hardcore German: Beim Kochen rühre ich die Suppe im Topf um. - While cooking, I stir the soup in the pot.

German often puts a time/place phrase first for emphasis. When you put something in position 1 (Beim Kochen), the finite verb must come next (position 2), so rühre comes before ich:

Beim Kochen (1) rühre (2) ich (3) …

This is the normal verb-second (V2) rule in main clauses.

What exactly is beim, and why is it used here?

Beim is a contraction of bei dem.

bei is a preposition that takes dative
dem is dative masculine/neuter singular (here: das Kochen → neuter)

So beim Kochen literally means at the cooking, and idiomatically while cooking / when cooking.

Why is Kochen capitalized? Isn’t it a verb?

Here kochen is used as a noun-like infinitive (a nominalized verb), so it’s capitalized: das Kochen = the cooking.
After bei + dative, it becomes bei dem Kochen → beim Kochen.

Could I also say Während ich koche or Wenn ich koche? What’s the difference?

Yes, and the meaning is very similar, but the feel changes:

Beim Kochen … = compact, typical for describing routines or instructions; focuses on the activity as a setting.
Während ich koche, … = explicitly during the time the cooking happens.
Wenn ich koche, … = more like whenever I cook (habitual), or sometimes if (depending on context).

Why is it rühre ich and not ich rühre?

Because Beim Kochen is in the first position, the verb must stay second. So you get rühre ich (inversion).
If you start with the subject, you get the more English-like order:

Ich rühre die Suppe im Topf um.

Both are correct.

Why does um appear at the very end of the sentence?

Because umrühren is a separable verb: um- + rühren.
In a main clause:

the conjugated part (rühre) goes in verb position 2
the separable prefix (um) goes to the end

So: rühre … um.

How would it look in a subordinate clause where the verb goes to the end?

Then the verb parts usually rejoin:

…, weil ich die Suppe im Topf umrühre. Here umrühre stays together at the end because subordinate clauses typically place the finite verb at the end.

Is there a difference between rühren and umrühren?

Often yes:

rühren = to stir / to move around (can be more general; also has other meanings like to move emotionally in different contexts)
umrühren = to stir around / thoroughly, the common choice for stirring food in a pot

In cooking instructions, umrühren is very common.

Why is it die Suppe (accusative)? How do I know it’s not dative?

die Suppe is the direct object of umrühren (what you are stirring), so it’s in the accusative:

Ich rühre (wen/was?) die Suppe um.

There’s no preposition before die Suppe, so it’s not a prepositional dative here.

Why is it im Topf and not in den Topf?

Because in can take either dative (location) or accusative (movement).

im Topf = in dem Topf (dative) → location: the soup is already in the pot
in den Topf (accusative) would suggest movement into the pot (e.g., pouring something in)

Here you’re stirring the soup that’s already inside, so im Topf fits.

Is im Topf necessary? Where can it go in the sentence?

It’s optional; it just adds detail (in the pot). You could say:

Beim Kochen rühre ich die Suppe um.

Word order is flexible, but common placements are:

Beim Kochen rühre ich die Suppe im Topf um. (as given)
Beim Kochen rühre ich im Topf die Suppe um. (possible, but often more marked/emphatic)

Typically, the object die Suppe stays close to the verb, and the separable um stays at the end.

What tense is rühre, and why is present used for something happening now?

It’s present tense (Präsens). German uses Präsens both for:

what is happening now
repeated/habitual actions
instructions or general statements

So it works well for describing a normal step you do while cooking.

How do I pronounce rühre and um?

rühre: the ü is like the French u in lune (lips rounded, tongue forward). The r varies by region; often a throat r in standard German.
um: short u like oo in book (not exactly English, but closer than oo in food).