Agreed. This should be a priority. If you keep illegals out at the border you don't have so many problems about identifying them within the borders.
Why amnesty? This makes no sense to me. Basically, you are rewarding people who have broken the law.(2) Use one of the Amnesty programs that works
I agree with you, though. Making the cost of hiring illegals higher than the cost of hiring legal labor would reduce the problems significantly. The problem there is that there have been numerous politicos who have been found to have hired illegals, either knowingly or not. Certainly, anyone who forges documents, for any reason, should be sent to prison, as well as being fined heavily. Simply make the risks so high that it becomes untenable.