Is it normal my dreams have moods?

Sure, dreams can affect your mood after waking up.
Very powerful dreams that involve family, closest friends, wife, or whatever else you deeply care about can affect your day.

I've had times where I had terrible dreams involving my parents. I had one about my mom being very ill and dying in hospital in front of me a few months ago. It was so real. Down to the call from my dad, me buying a plane ticket, renting a car, driving to the hospital. It felt real... After I woke up I felt devastated, I had a lump in my throat and was ready to start crying, even tho my mom was ok. I called her and talked to her on that day, so I had a full conformation she was ok, but I still felt like shit for most of the day. I was sad, felt tired and was generally very irritable for that day.

Most regular nightmares will wake you up and give you a scare, but you tend to ignore them cause they are nonsense. Monsters? Saw-type psychopaths? Falling off a cliff? Paranormal shit? Lovecraft horrors? You doing crazy stuff? Usually such dreams do not affect you, because your brain knows that dream was something that can never happen in the real world.

But take a really intense dream about a hard moment in your life (Family crisis, losing a loved one, cancer, other major diseases, etc...) and then turn it up to 11... That can affect you. And it can affect you deeply.
This ain't the terminator T-rex with a top hat you dreamed of that tore off your head. It's something like your mother dying in your arms and you being powerless to stop it.

Your brain is stuck on a really bad scenario that can happen in real life. Something you are usually not ready (or ever ready) to face. It all seems too real and you get all of the emotional baggage from just a stupid dream.

Naturally it's not just bad dreams... You could have a really intense one about winning at something, meeting your dream girl, making it big in life. Again things that are realistic and can happen. So again, your brain goes into overdrive, but now you feel great. You can take on the world! Positive intense dreams are the best thing ever, but sadly they are much rarer (in my case at least)

Very normal!

But it's not related to the dream. I had a dream about my friends getting attacked by pumas and I didn't really care after I woke up

Of course dreams affect your mood when you wake up. I once had a dream where I went to my bathroom, picked up a candle, candle wax dripped on my hands, and the wax started burning my arms off. In the dream, I actually felt pain. I woke up rather scared.