Ég er ósammála honum um hvort þetta starf sé gott fyrir hana.

Honum is the dative form of hann (he, him).

The adjective sammála / ósammála (in agreement / in disagreement) always takes its “person” in the dative case, not with a preposition:

• Ég er honum sammála. – I agree with him.
• Ég er henni ósammála. – I disagree with her.

So in your sentence:

• Ég er ósammála honum …
  literally: I am disagreeing-to him … → I disagree with him …

Using hann (accusative) here would be ungrammatical in standard Icelandic.

Ósammála is not a verb; it is an adjective (built from ó- “un-/dis-” + sammála “in agreement”).

So the structure is:

• Ég er ósammála honum.
  literally: I am disagreeing (adj.) to him.

Grammatically this is like saying:

• Ég er þreyttur. – I am tired.
• Ég er svangur. – I am hungry.

Just with the special property that (ó)sammála takes a dative complement (honum, henni, þeim etc.).

Both mean “I disagree with him”, and in most contexts they’re interchangeable.

Nuance:

• Ég er ósammála honum.
  Uses the negative prefix ó- in the adjective. Feels a bit more compact and slightly more neutral/standard.

• Ég er ekki sammála honum.
  Uses explicit negation (ekki) with the positive adjective. Can sound a bit more contrastive in some contexts (I’m not on his side in this), but often there’s no real difference in meaning.

Both are common and correct.

Um hvort here means “about whether”.

• um = about, regarding
• hvort = whether / if (in an indirect question)

So:

• …ósammála honum um hvort þetta starf sé gott fyrir hana.
  = …disagree with him about whether this job is good for her.

This pattern is very common:

• Við ræddum um hvort við ættum að flytja.
  We talked about whether we should move.
• Ég er ekki viss um hvort hann komi.
  I’m not sure whether he will come.

Very roughly:

• hvort = whether / if in questions (direct or indirect), or when there is a choice / uncertainty.
• ef = if in conditional sentences (“if X then Y”).

Use hvort:

• Ég veit ekki hvort hann kemur.
  I don’t know whether/if he is coming.
• Hún spurði hvort þú værir heima.
  She asked whether/if you were at home.

Use ef:

• Ef hann kemur, þá förum við.
  If he comes, then we’ll go.
• Ef þú lest meira, bætist tungumálið.
  If you read more, your language improves.

In your sentence there is an indirect question (we’re unsure whether the job is good for her), so hvort is correct:

• … um hvort þetta starf sé gott fyrir hana.
  … about whether this job is good for her.

Sé is the present subjunctive of vera (to be).
Indicative: ég er, þú ert, hann er …
Subjunctive: ég sé, þú sért, hann sé …

The subjunctive often appears in indirect questions and expressions of doubt, opinion, or uncertainty, especially after hvort.

Here the speaker is not stating a fact; they’re talking about a possibility / opinion:

• hvort þetta starf sé gott fyrir hana
  whether this job is (in fact) good for her

You can hear er in everyday speech, but sé is the more careful/standard choice in this kind of clause.

In subordinate clauses (introduced by hvort, að, þegar, ef, etc.), Icelandic does not normally use verb-second word order like in main clauses.

The most natural word order here is:

• hvort þetta starf sé gott fyrir hana
  [hvort] [subject] [verb] [complement]

You could say “hvort þetta starf gott sé fyrir hana” or “hvort þetta starf sé fyrir hana gott” in very marked, poetic, or highly emphatic styles, but in normal prose and speech:

• Subject – Verb – Rest is what you want in this kind of clause.
• Putting sé first (hvort sé þetta starf …) would usually sound wrong or at least very odd.

Hana is the accusative form of hún (she).

The preposition fyrir takes:

• accusative when it means “for (someone’s benefit)”, “for her / for him / for us”, etc.

So:

• fyrir hana – for her (benefit)
• fyrir hann – for him
• fyrir mig – for me

In your sentence:

• … gott fyrir hana.
  = (is) good for her → fyrir demands accusative, so hana is required.

You normally keep the um here.

The pattern is:

• ósammála e-m um e-ð – disagree with someone about something.

When the “something” is a clause, that clause can follow um directly:

• Ég er ósammála honum um hvort þetta starf sé gott fyrir hana.
  I disagree with him about whether this job is good for her.

Saying it without um:

• Ég er ósammála honum hvort þetta starf sé gott fyrir hana.

would sound weird or wrong to most speakers, because ósammála by itself just takes the person (dative), and um introduces the topic of disagreement.

No, that’s not idiomatic Icelandic.

The standard patterns are:

• Ég er honum ósammála.
• Ég er ósammála honum.
• Ég er ósammála honum um þetta. – I disagree with him about this.

Using við:

• ✗ Ég er ósammála við hann. (unnatural / incorrect)

The preposition við is used with other adjectives/verbs (e.g. reiður við hann – angry with him, tala við hann – talk to him), but (ó)sammála specifically takes a bare dative object, not við + accusative.