You've got quite a bit of confusion here.
1. Which is faster, parseInt(el.style.width) or el.offsetWidth?
2. Should I write parseInt(el.offsetWidth)?
No, offsetWidth returns a number. But you should specify the radix whenever you use parseInt!
3. Which is faster, getElementById() or childNodes[]?
If you got the parent use the childNodes. If you got the id, you can use byId. Performance will depend on the browser (test).
4. Which is faster, new Image() or appendChild('img')?
If you want to force the browser to download a resource, use new Image(). If you want to add an image to the layout use appendChild(IMAGE_ELEMENT)
5. Give me please a link to a splendid Javascript PERFORMANCE guide
I rather leave you with this link: http://jsperf.com/, and let you do A/B testing.
getElementbyIddoesn't traverse the DOM. It is an in memory lookup of the element. As such is extremely fast.