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Does music exist in the The Hunger Games universe? And if so, what kinds of music?

Rap, Electronic, Rock, Classical, Jazz, Street performers, etc?

Does music exist in Panem?

There is quite a bit of music interspersed throughout the series:

In Chapter Five of The Hunger Games there is music during the chariot procession:

The opening music begins. It’s easy to hear, blasted around
the Capitol.

In Chapter Ten there is music when they go on the roof the night before the Games start:

The roof is not lit at night, but as soon as my bare feel reach
its tiled surface I see his silhouette, black against the lights
that shine endlessly in the Capitol. There’s quite a commotion
going on down in the streets, music and singing and car horns,
none of which I could hear through the thick glass window
panels in my room.

In Chapter Eleven there is music when the casualties for the day are shown:

That’s it. The Capitol seal is
back with a final musical flourish. Then darkness and the
sounds of the forest resume.

In Chapter Sixteen Rue and Katniss talk about music, but it turns out that they're apparently talking about singing and not actual instrumental music:

Rue, who when you ask her what she loves
most in the world, replies, of all things, “Music.”

“Music?” I say. In our world, I rank music somewhere between hair ribbons and rainbows in terms of usefulness. At
least a rainbow gives you a tip about the weather. “You have a
lot of time for that?”

“We sing at home. At work, too. That’s why I love your pin,”
she says, pointing to the mockingjay that I’ve again forgotten
about.

“You have mockingjays?” I ask.

“Oh, yes. I have a few that are my special friends. We can
sing back and forth for hours. They carry messages for me,”
she says.

In Chapter Eighteen when Rue dies she asks Katniss to sing for her (which Katniss does) and Katniss reminisces about the music of her father singing:

“Sing,” she says, but I barely catch the word.

Sing? I think. Sing what? I do know a few songs. Believe it or
not, there was once music in my house, too. Music I helped
make. My father pulled me in with that remarkable voice —
but I haven’t sung much since he died. Except when Prim is
very sick. Then I sing her the same songs she liked as a baby.

We also see in that scene that Katniss actually had a music teacher:

What my music
teacher calls a mountain air.

In Chapter Twenty-Two we find out that they actually had music assemblies in school:

“So that day, in music assembly, the teacher asked who
knew the valley song. Your hand shot right up in the air. She
stood you up on a stool and had you sing it for us. And I swear,
every bird outside the windows fell silent,” Peeta says.

In Chapter Five of Catching Fire musicians play during the tour of District 11:

Somewhere below, musicians begin to play. As the first wave of our little procession
begins down the steps, Peeta and I join hands.

In Chapter Six musicians play at the party in President Snow's mansion:

About halfway between the floor and the
ceiling, musicians float on what look like fluffy white clouds, but I can't see what holds them
aloft.

Later at the party we find out that in District 12 they presumably had fiddle and flute music:

Music filters down from the clouds as he leads me away from the team, the table, and out
onto the floor. We know only a few dances at home, the kind that go with fiddle and flute
music and require a good deal of space. But Effie has shown us some that are popular in the Capitol.

In Chapter Twelve we find out that there is an electronic gadget called a music chip:

I begin to question them casually about what other hardships this winter has brought them.
They are not used to want, so any little disruption in supply makes an impact on them. By the
time I'm ready to be dressed, their complaints about the difficulty of getting different
products — from crabmeat to music chips to ribbons — has given me a sense of which
districts might actually be rebelling. Seafood from District 4. Electronic gadgets from District
3. And, of course, fabrics from District 8. The thought of such widespread rebellion has me
quivering with fear and excitement.

In Chapter Sixteen we get more information about what a music chip is:

“The strength of the thread,” Beetee finishes explaining. “Automatically. It rules out
human error.” Then he talks about his recent success creating a musical chip that's tiny
enough to be concealed in a flake of glitter but can hold hours of songs. I remember Octavia
talking about this during the wedding shoot, and I see a possible chance to allude to the
uprising.

In Chapter Two of Mockingjay music plays on television after Peeta's interview:

Music plays them out, and then there's a woman reading a list of expected shortages in the Capitol — fresh
fruit, solar batteries, soap.

In Chapter Sixteen there is music at Finninck's wedding:

The three hundred lucky guests culled from 13 and the many refugees wear their
everyday clothes, the decorations are made from autumn foliage, the music is provided by a choir of children
accompanied by the lone fiddler who made it out of 12 with his instrument.

This also tells us that there were multiple musicians back in District 12.

Yes. The Hunger Games Wiki has the following three:

Deep In The Meadow

"Deep In The Meadow" (also known as "Rue's Lullaby") is a song that is sung by Katniss to Rue, who was on her deathbed after Marvel speared her in the stomach. Rue's last request was to hear Katniss sing

The Hanging Tree

This song was taught to Katniss Everdeen by her father, Mr. Everdeen, when she was young. Her mother, Mrs. Everdeen heard her singing the song and watched Prim and her making necklaces of rope to go with it.

The Valley Song

The Valley Song is mentioned in the first Hunger Games book by Peeta in the caves, saying how he remembers Katniss singing this song at school.

All three of the songs are folk songs as befits people who lack the resources to make musical instruments. It is safe to assume that the Capital has its own music but Katniss never has much opportunity to experience it.