1) a sodium of 146 or 147...who cares? Absolutely not getting worked up about. As an intensivist, I wouldn't raise an eyebrow until consistently greater than 150...and not start thinking about treating until 155.
2) The difference between 146 and 147...that's likely well within the margin of error in the test, so again, it's a nonstarter for me. The rise after "treating" is meaningless.
3) Imagine you have a patient with real numbers that need attention. The key points are that unless you have a patient you've given massive amounts of NaCl to (say a TBI patient you're given 3% or 7% hypertonic saline to for hyperosmotic therapy), your hypernatremia is more a product of having a free water deficit. If you don't give back the free water you're not going to make much progress. NS has 154mEq of NaCl - and no free water. D10W is at the other end of the spectrum (need D10 to maintain appropriate osmolarity) and is completely free water. 1/2NS is 77 mEq NaCl and is exactly that: half free water. There are calculations you need to do to find the free water deficit. That calculation will tell you the total amount of free water you need to give above maintenance needs.
4) In cases where your Na is super high - like 170+, you will get some improvement with NS, but this should be thought of more due to expansion of intravascular volume not from giving free water back.
5) 1/2NS will frequently get you hyponatremia and there is plenty of evidence supporting that assertion in both adults and now, even in children. There are a growing number of institutions where adults can only get NS unless they have documented hypernatremia. For kids, the standard of using 1/4 or 1/3 NS is falling out of favor. The use of hypoosmolar fluids in peds is based on sodium needs per kg (2-4mEq/kg/d is what you'll find in most texts), but ignores that deviations from normonatremia are typically water problems, not salt issues.
6) In your hypothetical, again with the change to be numbers that are actually concerning, the most common error that would prevent from correction of the sodium as expected is not adding fluid in OVER the maintenance rate. If that's not done, you won't make any headway.