Okay, native Dutchie here, and I'll try to explain it as precise as I can (if you still don't understand, please say so, and I'll try again).
Uit is a location-based preposition. So if things come from somewhere (country, room, box, etc.), you will use uit. Like you did in your sentence 'Ik kom uit België'. (just an fyi, it's got to dots on the e in België, otherwise there could be trouble pronouncing it).
Van is a possesion-based preposition. So if someone or something owns something, you will use van. Like: Dat boek is van mij (That book is mine) (literally it says 'that book is of me'). You could also say 'Dat cadeau kwam van hem' (that present came from him), where you would translate it with from, like you asked about.
Personally, I would translate the English word out to Dutch as 'eruit', like in the sentence here (een kat komt eruit/a cat comes out). Another example would be a teacher saying 'Eruit!' to a student that's been perticularly annoying or whatever, and sends him out of class.
Just to make it more confusing, uit has another meaning (yes, I know, Dutch is horrible. No offense, but this is exactly one of the reasons why I don't get why you would want to learn a language that only about 25 million people speak, unless you are moving here). Uit is also the opposite of 'aan'. Which, again, has multiple meanings. 'Aan' and 'uit' can either refer to the status of a relationship (basically, aan means that there is a relationship, and if the 'relatie uit is', that means the couple broke up), or it refers to some kind of electronic thingy that you can turn on or off (on=aan, off=uit). Like 'ik zet de computer aan' (I'm turning the computer on), or 'De oven is uit' (the oven is turned off).
I hope this helped, and if it didn't, please ask again (and anyway, you helped me giving me a reason to procrastinate doing my homework ;)
Firstly, your effort swept me off my feet. Impressive! I'm grateful to your willingness to hang on even if I don't understand what you explained in the first place.I'm not moving to Holland and I find this language worthy of being learnt and one of the easiest out there('aan' and 'uit' do become pain in the neck but nothing's perfect) :)
Secondly, I am quite clear with 'uit' and 'van' now.Thanks to you.
Finally, the largest para is difficult for me to gulp down and I'm trying to find the little piece of logic,if it exists,, but I'll try to be patient and manage to live with it :D Thanks again!
With "heraus" the action is towards the speaker, with "hinaus" the action is away from the speaker. "Eruit" however does not imply any direction of action, and I'm not sure there are any words that do. I can only think of location such as "hieruit" and "daaruit", but again those don't indicate a direction of action.