Do you use this as a response to tak or do you use it to say someone is welcome (to come somewhere) or both?
In danish "du er velkommen" is NEVER used as a response to "tak". It is used in sentences like "du er velkommen til at besøge os" (you are welcome to visit us) or "du er velkommen til at tage et stykke kage mere" (you are welcome to take another slice of cake).
The most common response to "tak" is "det var så lidt" (it was nothing). "Velbekomme" is also a very common response to "tak" especially in relation to food. After a meal people say "tak for mad" and the person who made the food will answer "velbekomme" (velbekomme literally means something like "may it do you well"). A third common response to "tak" is "selv tak" (thank you yourself) which is very common in exchange situations where both parties gain something, like when you buy something at a store, etc.
As the sentence is written here, it is primarily used as a response to thank you. If I gave you a Lingot, you will tell me 'tak' (thank you/thanks) and I will say 'velbekomme/du er velkommen' (you are welcome).
Same word (velkommen) can also be used when welcoming people inside you house, but then more often put together with other words, such as 'velkommen til' or 'velkommen indenfor'
I hope it helps a bit
As a native danish speaker, I don't remember ever hearing this phrase used in response to 'tak'. That would sound awkward to me. For me, this is only used literally, when you want to actually welcome someone to the location or tell them that they would be welcome to come. In response to 'tak', it is more natural to say 'Det var så lidt', meaning something like 'It was not much (trouble)'
Not as a response to "Thank you", but yes, "you are" can be contracted to "you're".