Learn Icelandic: Ég hjálpa honum í dag.

The verb hjálpa (to help) always forces the person being helped into the dative case. While hann is the basic word for "he" (and the accusative form for "him"), its dative form is honum.

Would I use a different pronoun form if I were helping "her" or "them"?

Yes. Because hjálpa requires the dative case, you have to use the dative versions of those pronouns. You would say Ég hjálpa henni (I help her) or Ég hjálpa þeim (I help them).

Why does "today" translate to two words, í dag?

In Icelandic, "today" is literally expressed as "in day" (í dag). You will see this exact same pattern with other time expressions too, like í kvöld (tonight, literally "in evening").

What happens to the word order if I want to start the sentence with "Today"?

If you move Í dag to the beginning, the verb hjálpa must still be the second piece of information (this is a strict rule in Icelandic). The sentence becomes Í dag hjálpa ég honum.