1. "the bad side of the cut-off" probably means being someone who loses their job because the employer can't afford to pay them. Some workers will have their pay increased by the law, but some workers will lose their jobs because their employer can't pay them what is legally required (at least that is what is implied by the article). The workers who don't get higher pay but do get fired will be on "the bad side of the cut-off."
2. Good question. Washington is not a person is this instance. It is referring to the United States government, which is located in the city of Washington, D.C. Many Americans refer to the government as a whole in this way. For example, if someone says "Washington is so corrupt" they mean that the government in general is corrupt not necessarily specific persons.
3. A safety net is something meant to help people who drop below some level of income, test score, etc. Sometimes safety nets are used to prevent people from having even worse things happen to them. Unemployment insurance is an example of this.
4. An extension is something that prolongs or widens the time frame or numerical range. If I am buying a house from you and I need to wait for more money to buy it, I might ask for an extension on our deal. Similarly, if a business offers kids a discount and then changes their policy to include teenagers, that would be an extension of their age range for discounts.