Questions & Answers about Я надеваю шапку и шарф.
Why use надеваю instead of одеваю?
The verb надевать means to put an item of clothing on. The very similar verb одевать means to dress a person. Since you are putting on clothes and not dressing another person, надеваю is the correct choice here.
Why does the word for hat change to шапку?
The hat is the direct object of the verb (the thing being put on), so it takes the accusative case. For feminine nouns ending in -а like шапка, the ending changes to -у in the accusative.
If the scarf is also a direct object, why doesn't шарф change its ending?
Шарф is a masculine noun for a non-living thing (inanimate). In Russian, the accusative form for inanimate masculine nouns is exactly the same as their dictionary form, so it stays шарф.
The English translation says "my hat", but there is no word for "my" in Russian. Is it missing?
In Russian, it is completely normal to drop possessive words like мой or свой (my) when talking about your own clothing or body parts. It is already understood from the context that you are putting on your own clothes.