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> 喜んであなたのお手伝いをします。

is translated as

> I will be pleased to help you.

あなたのお手伝いを "help to you" and is the direct object of the verb します , I
understand this. But I don't understand why

  1. the verb _yorokobu_ is conjugated in a _te_ -form (which I understand is imperative like "be delighted!"), and

  2. it's translated as some sort of future passive voice since present passive voice usually ends with _-reru_ , but

  3. here it does not and it is translated in future tense (why is it "I will be pleased" and not "I'm pleased"?)
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translation て-form passive-voice
Why 喜んで in 喜んであなたのお手伝いをします。is in some sort of imperative voice and not in passive voice?
喜んで is just the 〜て form of 喜ぶ. And as you (might) know, one main usage of the
〜て form is to connect multiple clauses as "and":

>   * 喜びます → I am/will be glad.
>   * あなたのお手伝いをします → I will help you.  
>  ⬇︎
>   * 喜んであなたのお手伝いをします。 → I am pleased and will help you → I will be pleased to
> help you.
>

You could also think of this 〜て as the "by means of 〜" form (for which 〜で is
used on nouns).

>   * 電車でゴー！ → Let's go by train
>   * 魚を焼いて食べる → Grill a fish and eat it ("Eat a fish by grilling it").
>

So you could think of this sentence as `I will help you "by means of" being
happy`.

* * *

The form here is simply the 〜て form. There is nothing going on concerning the
passive voice (受身) or imperative (命令形). The passive form of 喜ぶ is 喜ばれる. The
imperative form is 喜べ, and while the 〜て form _can_ function as a "light"
imperative (as 〜てください, omitting the ください), that is not the case here. If the
sentence were _slightly_ changed to say something like the following, then
you'd have a "light" imperative.

>   * 喜んで **ね** 、あなたのお手伝いをします。 → Cheer up, I'll help you out.
>

But then the context of the sentence is somewhat different; you're comforting
the other person who almost seems bummed out about some large task and needs
help to get it done, whereas in the original sentence the person's feelings
about needing help feel more neutral.

* * *

Remember that the present and future tenses in Japanese are the same, and
although context can often help figure out which it should be, there are times
when it's ambiguous. In this case it's clearly the future because it would be
strange to say to someone, "I'm gladly helping you (now)!"
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answer
The origin of the form 喜んで as the _te_ -form of the verb 喜ぶ has been discussed
in @istrasci's answer. (I.e. it does _not_ express a request, but rather its
literal meaning is "I am pleased and will help you".)

The form 喜んで could be said to be slightly more than just the _te_ -form of
another verb. It is a very idiomatic expression that even has its own entry in
monolingual dictionaries, e.g. in 大辞林:

> **よろこんで** 【喜んで・悦んで】  
>  （ 副 ）
>
> 相手のことばを快く受け入れるさま。自分から進んで。快く。 「 －伺います」

where it is analyzed as an **adverb** , allowing for expressions such as

> はい、喜んで  
>  It will be my pleasure.
>
> 喜んで伺います  
>  I gladly [confirm your request].

You could interpret the form in your sentence as the plain _te_ -form of 喜ぶ,
or recognize it as a more idiomatic phrase adding a polite touch:

> 喜んであなたのお手伝いをします。  
>  It would be my pleasure to help you.
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answer