Eldivenlerimi takmadan önce ellerimi yıkamak zorundayım.

The combination verb stem + -madan + önce literally means “before doing [verb].” Here’s how it breaks down:

• tak = “put on”
• -madan = the adverbial form (sometimes called the “connegative”), which in this structure doesn’t negate but marks “before doing”
• önce = “before”
  So tak-madan önce = “before putting on.”

Eldivenlerimi carries three pieces of information:

1. -ler = plural (“gloves”)
2. -im = 1st person singular possessive (“my”)
3. -i = accusative case (marks a definite direct object)
   Altogether, eldiven-ler-im-i = “my gloves” as a specific, direct object.

Yes. Yıkamalıyım uses the necessity/modality suffix -malı/-meli (“should/must”), so ellerimi yıkamalıyım also means “I should wash my hands.” The nuance is subtle:

• zorundayım often feels a bit stronger or more official (“I’m obliged”)
• -malı/-meli is a bit more like giving advice or stating necessity in general.

Both verbs can mean “to wear,” but they have different uses:

• giymek is general for clothes, shoes, hats, etc.
• takmak is for accessories or things you “put on”/“attach,” like watches, glasses, jewelry—and yes, gloves. So native speakers normally say eldiven takmak.

Turkish is fairly flexible, but the most natural word order here is:

1. Adverbial clause: Eldivenlerimi takmadan önce
2. Direct object + infinitive + verb: ellerimi yıkamak zorundayım
   You could shift small phrases for emphasis, but verbs typically stay at the end, and time/place clauses usually come early. A totally scrambled order would sound odd.