I'll go ahead and get this out of the way: you can request soldiers all willy-nilly, but no two can be the same. XCOM:EU uses the following soldier format:
First name: Maximum 11 characters.
Last name: Maximum 15 characters.
Nickname: Maximum 11 characters; only available at Sergeant rank or better (3rd rank up)
Class: Assault (Fast and short-range), Support (Medics and riflemen), Heavy (Machineguns and explosives), and Sniper (Long-range and scouts) classes are assigned randomly at the first rank up. If you want a specific rank, you'll have to wait until a Squaddie with the proper class pops up.
Gender: Assigned randomly at soldier creation. Non-adjustable. There seem to be fewer women than men.
Nationality: Assigned randomly at creation. Non-adjustable. Your guy gets a little flag patch on his back, and a similar flag on his profile, and nothin' else. Not even an accent! For shame. Don't know what all countries are represented, on account of me not knowing flags... at all. I can recognize a few though. AMURICAH
Race: A few options for general racial type.
Voice: Several different voices. They all say the same quips during combat though, just sliiiightly differently.
Head: A number of general face shapes.
Skin Color: Based on Race, gives you a few different sub-options.
Hair/Helmet: Men get short hair, ladies get a few options for ponytails and the like, either can choose one of three different helmets... or the amazing original X-Com haircuts.
Hair Color: Many neon options for hair color, and a few sensible ones.
Facial Hair: Mustaches and beards, sideburns and mutton chops, for dudes only. Sorry ladies.
Armor Deco: Most types of armor have two different options for their appearance; one usually has more color than the other, which tends to be taken up by grey-green plates. Usually.
Armor Tint: Fairly large selection of colors for armor. Useful for telling dudes apart in the chaos of battle. Most basic colors are represented, and a few combos that are opposite each other on the color wheel. Red/green, purple/yellow, blue/orange, etc. Default is a drab olive color.
Specify what you'd like in the stream or on here and I'll try to make it happen.
IF YOU DIE, YOU'RE DEAD. You go up on the memorial wall, and can not be 'reborn' through identically naming another rookie. You can always request a differently named soldier, though.