Question about some parts of this YouTube video (Adele on talk show)

I can’t personally watch the video but a triumph is a success and I would think that up numbers are songs which are upbeat and fast rather than the slow ballads that she often sings. No idea about the other questions without watching it

“I got a new coat out of the budget today” means that a new coat was bought for the production and paid for out of the production budget, and she gets to keep the coat.

37s He says I’m loving this bob. He likes here new hair style. Its much more manageable.
51 Hair lines -She is referring to hair extension or clip ins. Rather than a wig.

1316 He says she needs a hype man. – Someone who cheers on the crowds to get them to participate “during the up numbers” the songs that are upbeat and exciting and get people to move.

It’s a triumph, means it’s a success. It’s a winner. He’s complimenting her on her new hairstyle.

I’m going to use my hairline. I think she means she likes her hairline. She wants to highlight it. Make it a more prominent feature. And she’s going to use a wig (or perhaps a weave) to do that. I’m a bit unclear on the meaning here myself.

“I got a new coat out of the budget today”. The video would have had a budget. An amount of money that could be spent on it. The spending included a new coat for Adele. And she just poured coffee on her new coat.

“I think you need a hype man”, is as @LuckyGuy suggests someone to wind up the crowd. Like they have off camera in game shows. Someone to get the crowd going and get them participating.