How can I cut a section in a svg to an new file

Also, there is an Export Selection command in the File menu.

I would open the file in illustrator and just select what you need. It should be straightforward unless there is something else going on with the file.

The question is a little vague. If you're trying to extract elements out of an existing piece of artwork there are multiple ways to do that. But it depends on the artwork itself. It may be as simple as ungrouping the objects (if they're grouped) and then selecting the desired pieces of artwork, copying them to the clipboard and then pasting them into a new Illustrator document. If you're trying to cut portions out of existing shapes the Pathfinder tools can do that. Clipping masks can show/hide portions of artwork non-destructively.

If the artwork is vector-based already (given that it is in a SVG file) then there is no need for exporting in PNG format and then somehow going back to SVG. PNG is a pixel-based format. Converting vector shapes to pixels and then converting back into vectors again doesn't make any sense. It would degrade the artwork.