Is this normal behaviour in a partner?

My female friend asked me to ask on here about her boyfriend's behaviour, she's really upset and I think she may or may not be in an abusive relationship. She's worried and so am I.

Here is what she wrote to me in an email to say on here:

"We went to a pub (known as bar in the US) for lunch, he asked me specifically to look after his MP3 player so I put it in my bag to keep it safe, he went inside the pub toilet with my bag and as soon as he came out, he freaked out at me asking "Why is my MP3 player in your bag?", I reminded him "You asked me to look after it", I was more shocked he searched through my bag.

Then later when I stayed over at his flat (again in the US they called it something else, apartment I think), he went out and blowed his £10.00 on energy drinks and sweets then later he texted his friend that apparently £5.00 went missing so again he accused me even though I gave him food after he wasted his last money on getting drunk with his friends.

Then I woke up in his flat one day catching him on my phone using my internet and ringing people without asking me first. He also threatened to hurt my best male friend just because he changed my password on my laptop by mistake (he thought he was using his account on the laptop).

And he's obsessed with setting things on fire, he asked me to hold a piece of A4 paper up so he could use my lighter and his deodorant to burn the paper. I was scared and confused then later he confessed to me him & his friend stole an elderly woman's purse and used her money for then later rung her up on a payphone telling her where it is but set fire to it. Then he told me he's wanted by the RSPCA because he was filmed on someone's phone stabbing a dead cat's body with a gardening fork. He told me the cat was already dead but when I asked him to look after my kitten whilst my flat was being cleaned out, he kept making jokes lying to me telling me my kitten is dead.

Also he told me wouldn't sleep with me unless I lost weight but he happily asked for BJs."

So what do you think? Is my female friend in a healthy relationship?