I, too, am not a statistician but have to work with stats routinely. The concept of a “margin of error” being the inverse of(sqrtX) makes “margin of error” a statistical construct rather than an indication of the real potential for the estimate to be incorrect. The underlying thing being measured is one of the factors influencing true likelihood of the estimate being incorrect. For example, if I am surveying 1000 people regarding their eye color or sex, I will get a more estimate than if I am asking them whether Istanbul has a larger population than Athens or if their spouse trusts the employee at the checkout counter. Some things are inherently more difficult to estimate, are complex in detail, or are subject to change. A more professional measurement of the accuracy of an estimate of an attitude such as this is reliability, measured in several ways, or validity, much more difficult to establish and also measured in a variety of ways.
Only if the 1000 are truly “random.” In most opinion polls, they are not.