Learn Dutch: Dag meneer, welkom! - Hello sir, welcome!

You are right! The literal translation of dag is "day". However, in Dutch it is widely used as a short, friendly greeting meaning "hello" or "good day". Interestingly, you can also use dag to say "goodbye".

What word would I use instead of meneer to welcome a woman?

Meneer is the polite way to say "sir" or "Mr.". If you want to politely welcome a woman, you would use mevrouw (madam or Mrs.). The sentence would then be: Dag mevrouw, welkom!

Can I use welkom to say "you're welcome" after someone says thank you?

No, welkom is only used as a greeting when someone arrives at a place, just like in this sentence. If someone says "thank you" and you want to reply "you're welcome", you should use the phrase graag gedaan.