Earl hit the reply button quicker than I, and he's right. The answers to your questions are best answered by a person educated in the quality sciences. Exact answers depend on your industry and desired quality level. These requirements will be different for components critical to keeping a satellite operating over the course of years compared to the little things that play music from greeting cards.

I've answered your questions below, but notice that the responsibility is placed on you. Your responsibility is also governed by print, contract, and regulatory requirements.

You must perform those activities (inspection and testing) necessary to ensure that received materials meet your requirements.
2. If so how do we decide what to inspect?

Percentage sampling is not recommended. The best sampling plans have tables that specify the sample size and number of allowable rejects within the sample. One of my preferences is the C=0 table found in IPC-6012. In 6012, varying AQL levels are specificed for critical, major and minor defects. If no defects are allowable, 100% inspection is necessary (very expensive when properly done). Companies also have suppliers certified to supply parts of acceptable quality without incoming inspection. The qualification and monitoring efforts needed to maintain such a system are beyond the scope of this response, but Earl gives you a flavor of those requirements.
4. Are there any specifications/documents recommended by IPC?

IPC-6011, IPC-6012 and IPC-A-600 are the beginning points for rigid bare boards. I don't recall the written spec number, but IPC-A 610 is a beginning point for assemblies.

Good luck with you rendeavor. Remember that an effective quality system integrates incoming inspection with the rest of the company's efforts to provide customer satisfaction.