Stuðningurinn frá fjölskyldunni minni gaf mér líka meiri virðingu fyrir sjálfri mér.

The subject is Stuðningurinn frá fjölskyldunni minni.

The core subject noun is stuðningurinn = the support.
The phrase frá fjölskyldunni minni = from my family just tells you where that support comes from.

A useful clue is case:

• stuðningurinn is nominative, which is the normal subject case
• fjölskyldunni is dative because it comes after frá

So fjölskyldunni cannot be the subject here.

Because -inn is the suffixed definite article in Icelandic.

• stuðningur = support
• stuðningurinn = the support

Icelandic usually adds the to the end of the noun instead of using a separate word like English does.

Here it is definite because the sentence is talking about a specific support: the support that came from the speaker’s family.

Because the preposition frá always takes the dative case.

So:

• dictionary form: fjölskylda = family
• after frá: fjölskyldu
• with the definite article: fjölskyldunni

The possessive minni also has to match the noun in gender, number, and case:

• fjölskylda is feminine singular
• after frá, it is dative singular
• so minn becomes minni

That is why you get frá fjölskyldunni minni.

That is very normal in Icelandic.

Possessives like minn, þinn, hans often come after the noun, especially in ordinary, neutral phrasing:

• fjölskyldan mín = my family
• fjölskyldunni minni = to/from my family depending on the preposition or verb

Putting the possessive before the noun is possible in some contexts, but it is often more emphatic, contrastive, or stylistically marked. For a learner, the safest default is that noun + possessive is very common.

Because the verb gefa follows the pattern:

einhver gefur einhverjum eitthvað
= someone gives someone something

So with gaf:

• mér is the person receiving something, so it is in the dative
• virðingu is the thing given, so it is in the accusative

This is one of the most common verb patterns in Icelandic:

• Hann gaf mér bók = He gave me a book
• Stuðningurinn gaf mér meiri virðingu = The support gave me more respect

Because virðingu is the direct object of gaf, and direct objects are often in the accusative.

The noun is:

• nominative: virðing
• accusative: virðingu

So in this sentence:

• meiri virðingu = more respect

If virðing were the subject, then you would expect nominative virðing, but here it is something that was given, so accusative virðingu is correct.

Because Icelandic normally says virðing fyrir einhverjum/einhverju for respect for someone/something.

So this is a set pattern:

• virðing fyrir mér = respect for me
• virðing fyrir sjálfri mér = respect for myself

This is something worth learning as a chunk:
að bera/hafa virðingu fyrir ... or simply virðing fyrir ...

Even though English uses for here too, you still have to learn the Icelandic case pattern, because fyrir in this expression takes the dative.

Because sjálfri mér means myself, not just me.

Compare:

• fyrir mér = for me
• fyrir sjálfri mér = for myself

Icelandic forms this idea with:

• a pronoun: mér
• plus a form of sjálfur: sjálfri

So the sentence is specifically saying respect for myself, not just respect for me.

Also, Icelandic does not use sig for first person.
For myself, you use forms like:

• mig sjálfa / sjálfan
• mér sjálfri / sjálfum

depending on case and gender.

Yes. Sjálfri shows that the speaker is feminine.

The word sjálfur changes for gender:

• feminine dative singular: sjálfri
• masculine dative singular: sjálfum

So:

• fyrir sjálfri mér suggests a female speaker
• a male speaker would normally say fyrir sjálfum mér

This is a very useful detail, because Icelandic often shows the speaker’s gender in forms like this.

Because sjálfur works like an adjective, so it normally goes together with a pronoun or noun.

You can think of it as:

• mér = me
• sjálfri mér = myself

The self idea is not usually expressed by sjálfur alone. It needs something to refer to.

That is why Icelandic says things like:

• ég sjálf = I myself
• mig sjálfa = myself in the accusative
• mér sjálfri or sjálfri mér = myself in the dative

Somewhat, but not completely.

The sentence has normal Icelandic main-clause structure:

• first element: Stuðningurinn frá fjölskyldunni minni
• second element, the finite verb: gaf

That follows the common Icelandic V2 pattern, where the finite verb comes in second position.

As for líka, its position is fairly natural here:

• gaf mér líka meiri virðingu ...

Here it means the support also gave the speaker more self-respect.

You can move adverbs in Icelandic for emphasis, but not freely in every direction. The given word order sounds natural and neutral. For a learner, it is a good model to follow.