What happens if European commits a serious crime in US soil?

They are arrested and charged with the crime in the jurisdiction where the crime occurred. If found guilty they would serve time in the same prison as would anyone else convicted of that crime in that court. After they had completed their sentence then they would be deported.

If the crime was eligible for the death penalty then yes they could be sentenced to death.

Probably same as here. You would be arrested. They would tell your embisay. From there it would go to hearing. A Judge would decide that. Probably prison.

I was the jury foreman on a trial between a man and his wife - both were Somali.
He attacked her with a knife - trial was slow because of the need for interpreters for both parties.
We found him guilty, and he was sentenced to three years in prison, followed by deportation.

Exactly the same as happens to an American. It happened in the US, and US law and due process applies, up to and including the possibility of execution if they committed first degree murder in a state that uses the death penalty.

The only difference is that after finishing their sentence, they will be deported and no doubt banned from entering the US again. Or for all I know, the US might have a scheme for deporting foreign prisoners after part of the sentence. The first thing this person should do is ask to contact their embassy, who can tell family at home what's happened, but even the embassy can't do anything about justice taking its proper course. I've read the leaflet that British Embassies provide and all that says is that they can recommend a good lawyer, they can come and visit you so you've got a visitor, but that's about it.

It's the same everywhere - your country cannot interfere in the proper process of law of another country.

they are charged with the crime, and yes in some states they would get the death penalty
In Texas it is common for illegal aliens from Mexico to be sentenced to prison and death.

You're always subject to the laws and punishments of the country you're in unless you have diplomatic immunity, and that's fairly restricted.

European countries and Canada will refuse to extradite people to the USA if that person faces the death penalty.