ALL THE RIGHT FRIENDS IN ALL THE RIGHT PLACES= The subject and verb of this sentence appears two lines later: They got. If you have all the right friends in all the right places, you know important people who can help you. In this case, he's talking about the music industry.

THEY GOT= (coll.) The've got / They have

WE'RE GOING DOWN= In this context this is not talking about the present, but about the future, because now they are famous but they think they will lose popularity in the future because they haven't got the right friends or the right faces that you need to succeed today. So now they're famous, but in the future they'll go down (lose fame).

THEY GOT= (coll. AmE) They've got / They have.

ALL THE RIGHT MOVES= They can move and dance in style.

ALL THE RIGHT FACES= If they have the right faces they are beautiful. Today, most famous singers are beautiful and fashionable because image has become more important than the quality of their music. That's what they are criticising in this song.

PAINT THE PICTURE OF...= Imagine...
We can also use the verb "to picture" with the meaning of "imagine", "have a mental image":
- Can you picture this car in green? = Can you imagine this car in a green colour?

THEY GOT IT BETTER...= Things are easier for them.

YA= /jə/ a colloquial pronunciation of YOU (quite common in conversational speech)

...KING OF HEARTS...QUEEN OF SPADES= These are two playing cards, but they signify something in particular here, since they are confronted. In a card game, kings are more important than queens, so a king card wins over a queen card. A king of hearts would be a charming person who wins people's hearts (because they got all the right moves and all the right faces). The queen of spades might be a reference to Alexander Pushkin's story, "The Queen of Spades", in which the hero loses everything when he bets on the queen of spades in a gambling game. Anyway, playing cards the king is the winner, so One Republic says they'll help you (queen of spades) to fight the "king of hearts". They will try to give you good music to stop all those new prefab artificial singers that are dominating the music market.
The 4 suits of a pack of cards are: hearts ♥, spades ♠, diamonds ♦ and clubs ♣.

WE'LL FIGHT FOR YOU LIKE WE WERE YOUR SOLDIERS= This use of LIKE is colloquial. In standard British English we use LIKE (preposition) for comparisons with nouns, and AS IF (conjunction) for comparisons with verbs. So this sentence would be: We'll fight for you as if we were your soldiers. The verb WERE is not a past tense, it's an "unreal past", which expresses a hypothetical situation, an unreal situation: we are not your soldiers, but we'll fight for you the same as soldiers would. In fact, the grammar of this sentence is the equivalent of a conditional close: if we were your soldiers we would fight for you (but we aren't), but the meaning is: we will be your soldiers and we'll fight for you.

WE GOT IT GOOD= We are successful, we're doing fine.

THEY GOT IT MADE= They don't need to work hard to succeed because all their important friends are helping them.

THEIR GRASS IS GETTING GREENER EACH DAY= Things are getting better and better for them. This idea comes from the common expression "grass is always greener on the other side of the fence", which means that we always think that other people's circumstances are better than ours. So the green grass is here a metaphor for "good circumstances", luck. The greener the grass is, the better things are for them. The word THEIR here sounds /ðə/ (exactly the same as THE), which is a very common weak form for this word inside a sentence. We know it's THEIR and not THE because of the context (if things are better for them, it's THEIR grass which is getting greener, not grass in general).

THINGS ARE LOOKING UP= Now we have a good situation.

BUT THEY'LL TAKE US DOWN BEFORE ANYBODY'S KNOWING OUR NAME= But those new prefab singers will succeed and leave no room for us, so we won't be able to compete with them and people will forget us, we will no longer be famous.

AM I BRIGHT ENOUGH TO SHINE IN YOUR SPACES= My music is good enough to be with you?

THE NOISE YOU HEAR= Bad music.

I CAN'T BE POSSIBLE THAT RAIN CAN FALL ONLY WHEN IT'S OVER OUR HEADS= I can't believe how unlucky we are. (when there is a cloud, it only rains on us, not on the rest of the people)

IT DON'T MATTER= Of course it should be "it doesn't", but in colloquial English (both AmE and BrE) this can happen.

THAT'LL= Notice the pronunciation /ðætəl/