Learn Hardcore German: Jedenfalls komme ich pünktlich, auch wenn es regnet. - In any case I will arrive on time, even if it rains.

Because of German’s verb-second (V2) rule in main clauses. When you start with something other than the subject (here Jedenfalls), the finite verb must still be in second position, so the subject follows it:

Jedenfalls komme ich pünktlich, ... If you start with the subject, you get:
Ich komme pünktlich, ...

What does jedenfalls mean, and how is it different from auf jeden Fall or in jedem Fall?

jedenfalls = in any case/anyway; often a discourse marker to sum up or steer back to a point. Slightly less strong than a firm promise.
auf jeden Fall = definitely/for sure; stronger commitment.
in jedem Fall = in every case; more formal. Examples:
Jedenfalls komme ich pünktlich. (In any case, I’ll be on time.)
Auf jeden Fall komme ich pünktlich. (I’ll definitely be on time.)
Ich komme in jedem Fall pünktlich. (More formal.)

Where can jedenfalls go in the sentence? Do I need a comma after it?

Common placements:

At the start: Jedenfalls komme ich pünktlich, ...
In the middle: Ich komme jedenfalls pünktlich, ... You normally do not put a comma after Jedenfalls when it begins the sentence.

Why is komme (present tense) used to talk about the future?

German often uses the present tense for future events, especially when context makes the time clear. Ich komme can mean I’ll come. You can also use the future:

Ich werde pünktlich kommen, ... (slightly more formal or emphatic about futurity)

What exactly does auch wenn mean? Is it the same as wenn, obwohl, or selbst wenn?

wenn = if/when (plain condition or time)
auch wenn = even if (concessive: regardless of the condition)
selbst wenn = even if (often a bit stronger/more hypothetical)
obwohl = although/even though (concessive with a fact) Compare:
Ich komme pünktlich, wenn es regnet. (if it rains; plain condition)
Ich komme pünktlich, auch wenn es regnet. (even if it rains; rain won’t stop me)
Ich komme pünktlich, selbst wenn es regnet. (even if; stronger)
Ich komme pünktlich, obwohl es regnet. (although it is raining)

Why is there a comma before auch wenn es regnet?

German requires a comma before a subordinate clause. auch wenn es regnet is a subordinate clause, and its verb goes to the end:

..., auch wenn es regnet.

Can the auch wenn clause come first?

Yes:

Auch wenn es regnet, komme ich jedenfalls pünktlich. When a subordinate clause comes first, it occupies the first position, and the verb of the following main clause still appears in second position: ..., komme ich ...

Do we really need the auch? What’s the difference between wenn and auch wenn here?

wenn alone gives a plain condition (if).
auch wenn gives a concessive meaning (even if, regardless). If you drop auch, you often add trotzdem to keep the concessive sense:
Ich komme trotzdem pünktlich, wenn es regnet.

Is pünktlich an adverb or an adjective here? Where can it go?

Here it functions adverbially, modifying komme (on time). German uses the same form for adjectives and adverbs. Natural options:

Ich komme pünktlich.
Ich komme pünktlich an. (with the separable verb ankommen; explicitly “arrive on time”)
Ich bin pünktlich. (I am on time/punctual)

Could I say Ich komme pünktlich an instead of Ich komme pünktlich?

Yes. Ich komme pünktlich is perfectly idiomatic and understood as “I’ll arrive on time.” Ich komme pünktlich an uses the verb that explicitly means “arrive” and is very clear; both are fine.

Why do we need the subject ich? Doesn’t the verb ending already show the person?

Standard German normally requires an explicit subject pronoun. Even though the verb ending shows person/number, you still say ich here.

What is the role of es in es regnet?

It’s a dummy (expletive) subject used in weather expressions. It doesn’t refer to anything. Standard usage keeps it:

es regnet, es schneit, es hagelt In casual speech you may hear contractions like wenn’s for wenn es:
Auch wenn’s regnet, ... (informal)

Are there stylistic variants for the concessive clause?

Yes:

Wenn es auch regnet, ... (somewhat formal/literary; “even if/though it rains”)
Egal, ob es regnet, ... (no matter whether it rains)
Selbst wenn es regnet, ... (even if; stronger emphasis)

Why not auch ob es regnet?

Because ob introduces an indirect yes/no question (whether), not a conditional or concessive clause. auch ob is ungrammatical here. Use auch wenn, selbst wenn, or obwohl depending on the meaning.

Any spelling tips for words in this sentence?

jedenfalls is one word, all lowercase. Do not write jeden Falls.
auf jeden Fall (if you use that variant) is three separate words.
pünktlich has an umlaut. If you can’t type ü, write puenktlich.

Could I write the sentence with the future tense? Is the word order correct?

Yes:

Jedenfalls werde ich pünktlich kommen, auch wenn es regnet. The finite verb werde is in second position, and the main verb kommen goes to the end of the main clause. The subordinate clause stays the same.