Learn Hardcore Korean: je bang chimdae yeope begaewa iburi nohyeo isseoyo. - Next to the bed in my room, there are a pillow and a blanket.

제 is the humble/polite form of 내, coming from 저의. A native speaker uses 제 when talking politely or to someone older. 내 is informal, and 나의 is grammatically correct but rarely used in everyday speech.

Why is there no particle between 제 방 and 침대 옆에?

This is a stacked noun phrase. It really means “in my room (there is) the bed’s side.” You could expand it to 제 방에 있는 침대 옆에, but Koreans often compress relative clauses, dropping 에 있는 when it’s clear from context.

Why is 에 used after 침대 옆?

옆 is a location noun (“side”), so attaching 에 marks the specific place where something exists. 침대 옆에 = “at the side of the bed.”

What’s the role of 와 in 베개와 이불? Could I use another particle?

와 means “and” after a noun ending in a vowel. You could also say 베개하고 이불 (more casual) or 베개랑 이불 (colloquial), but 와 is slightly more formal than 하고 and neutral in tone.

Why do we say 놓여 있어요 instead of just 놓아요 or 두어요?

놓여 있다 is a passive–stative construction describing that something has been placed and remains there.

놓아요/두어요 describes the action of putting something down.
놓여 있어요 describes the resulting state: “it is placed.”

Could I simplify the sentence to 베개와 이불이 있어요?

You could, but that only tells someone the pillow and blanket exist. 놓여 있어요 specifically conveys that they’re lying there, ready to use, emphasizing their location and placement.

What is the dictionary form of 놓여 있어요?

The base verb is 놓이다, the passive of 놓다 (to put). In present progressive/stative form it becomes 놓여 있다, and politely 놓여 있어요.

Can I switch word order, for example, say 베개와 이불이 제 방 침대 옆에 놓여 있어요?

Yes. Korean is flexible with topic/object order. Both

제 방 침대 옆에 베개와 이불이 놓여 있어요
베개와 이불이 제 방 침대 옆에 놓여 있어요
are correct and mean the same. You choose based on what you want to emphasize first.