Learn Russian: Я хотел бы этот номер, потому что там красивый вид. - I would like this room, because the view there is beautiful.

Using хотел бы (I would like) makes your request polite and soft. If you just use хочу (I want), it can sound too direct or demanding when booking a room or ordering at a restaurant.

If a woman is saying this sentence, how does it change?

A female speaker would say Я хотела бы. To make a polite request with бы, we use the past tense form of the verb, which must always match the gender of the person speaking. The particle бы stays exactly the same.

What case is этот номер (this room) in?

It is in the Accusative case because it is the direct object - the thing you are requesting. Because номер is a masculine, inanimate noun (not a living thing), its Accusative form looks exactly the same as its dictionary form.

Why does the sentence use номер for room instead of комната?

In Russian, номер is used specifically for a hotel room. If you are talking about a room in a house or an apartment, you would use the word комната.