Glugginn opnast ekki, því hann er fastur, og ég þarf að ýta varlega.

Glugginn means the window (definite). Icelandic typically marks definiteness with a suffix attached to the noun:

• gluggi = a window
• glugginn = the window
  Here -inn is the masculine nominative singular definite ending.

Opnast is the so-called middle voice form (often used like “open” intransitively): it means opens / gets opened (i.e., becomes open), without naming an agent.

• Ég opna gluggann = I open the window (transitive; I do it)
• Glugginn opnast = The window opens (intransitive; it happens)

In main clauses, ekki usually comes after the finite verb (the verb that’s conjugated):

• opnast (finite verb) + ekki (negation)
  So Glugginn opnast ekki = The window doesn’t open.

Því here functions like because / since, introducing the reason. It’s a common, slightly formal/neutral way to give an explanation:

• ..., því hann er fastur = ..., because it is stuck.
  You’ll also see alternatives like af því að (very common, often a bit more explicit/colloquial).

Yes. After því introducing a reason clause, you normally get subordinate-clause word order, where the finite verb typically comes after the subject (and often after certain adverbs):

• hann er fastur (subject hann
  • verb er)
    This looks the same as English here, but in many Icelandic subordinate clauses you’ll notice differences especially when adverbs like ekki are involved.

Icelandic pronouns follow grammatical gender, not natural gender. Gluggi is masculine, so you refer to it with hann (masc. “it/he”).

• masculine: hann
• feminine: hún
• neuter: það
  In English we’d usually say it, but Icelandic uses the gendered pronoun.

Fastur is an adjective agreeing with glugginn (masculine, singular, nominative). Adjectives change form to match the noun’s gender, number, and case:

• masc. nom. sg.: fastur
• fem. nom. sg.: föst
• neut. nom. sg.: fast
  So with glugginn (masc.), you get fastur.

Þurfa commonly takes an infinitive clause introduced by að:

• ég þarf að ýta = I need to push
  It works a lot like English need to + verb, except Icelandic uses að
  • infinitive.

Þurfa mainly means to need / to have to. Depending on context, it can sound like must/have to rather than a soft “need”:

• Ég þarf að fara = I need to go / I have to go
  In your sentence it’s a practical necessity: you need to push gently.

Ýta means to push. You can use it without stating the object if it’s obvious (here: the window). If you do include the object, it typically takes the accusative:

• Ég ýti gluggann = I push the window
  You can also use prepositions to specify direction, e.g. ýta á (push on/press).

Yes, several natural alternatives exist, for example:

• Glugginn opnast ekki af því að hann er fastur, svo ég þarf að ýta varlega.
• Glugginn opnast ekki. Hann er fastur, svo ég þarf að ýta varlega.
  These keep the meaning but change the connector style (því / af því að / svo) and the sentence structure.