does иду work here?
No, because it implies going in a single direction, while "walking near" suggests going back and forth (or at least direction is not important in the statement).
Why is it "krovati" and not just "krovat" since it's a singular object?
It's genetive because of возле
why is "иду" incorrect?
Is У correct here instead of ВОЗЛЕ ?
Does близко work?
It does not, and I would not use it even if you made in grammatical.
Why иду isn't correct?
Why is постел wrong?
walking during sleeping !