5. There is no interdependence among individuals or firms other than what is channeled through the market. Everything else is a matter of simply adding up: me plus you plus her plus Microsoft plus the local taco truck. We live in isolated worlds, only connecting through the effects of our choices on the prices others face in the market. This assumption denies all the other ways we affect one another; it is very eighteenth century. In all other social sciences it has disappeared, but in economics it rules. Yet non-market interconnections matter mightily at all levels. There are discourses in asset markets, for instance: different narratives that compete and draw or lose strength from the choices made by market participants and the reasons they give for them. Consumption choices are profoundly affected by the consumption of others—this is what consumption norms are about. The list of interactions, of what makes us members of a society and not just isolated individuals, is as long as you care to make it. Technically, the result is indeterminacy---multiple equilibria and path dependence—as well as the inability to interpret collective outcomes normatively (“social optima”). Also, such models quickly become intractable.
This is a major point that frustrates me with economics. In political science there is a lot of open derision about this point, though these assumptions can be useful for specific purposes and continue to be used by certain narrow approaches (most notably, public choice approaches). As Peter Dorman says, it's an 18th century idea that has long been superseded by further research. It's not just wrong, we know its wrong. Yet, economists still use it and don't seem to acknowledge this.
When choosing your starting assumptions (assumptions must be made for any real analysis), it's essential to choose those assumptions based on your best read of the available evidence. Since the study of human psychology is outside the realm of economics they should be drawing their behavioral assumptions from other disciplines, like psychology and neuro-biology. This is happening to a certain extant but what needs to happen is that the assumption of individual, autonomous units is ejected wholesale as it has been done in other disciplines.
6. Rational decision-making takes the form of maximizing utility. Utility maximization is a representation of rational decision-making in a world in which only outcomes count and all outcomes are commensurable. This is another eighteenth century touch, this time at the level of individual psychology. Today almost no one believes this except economists. It is not a matter of whether people are rational, by the way, but what the meaning and role of rationality is. It is not irrational to care about processes as well as outcomes, or to recognize that some outcomes do not trade off against others to produce only a net result. (This is what it means to be torn.) The utilitarian framework is a big problem in a lot of applied micro areas; it is less clear what damage it does in macro. I suspect it plays a role in wage-setting and perhaps price setting in contexts where long-term supplier-customer relationships are established. We have a few models from the likes of George Akerlof that begin to get at these mechanisms, but their utilitarian scaffolding remains a constraint.Another area where economists are badly out of date. People care about more than outcomes and people do more than maximize. How you get there can be as important as where you're going. Another well established point often ignored by economists. Other social sciences embraced this notion decades ago, economists should catch up (and some are doing so).
Looking over the comments on this post I also note how angry people get by these assertions. The fact that some economists are taking other approaches is used to forgive those who don't, which is absurd. Also, the mere mention of these problems lead to frothing at the mouth about the government. It's of course by no means certain that the government is an ideal solution, but it is true that these bad assumptions have the disturbing quality of always leading to an answer that government is the problem (this simply leads people who believe in the scientific method to doubt their anti-government claims since there is such a strong relationship between the pseudo-science approach and the answer of government is bad)*. This is ideological blindness at its worst, people like the results given by these assumptions so they assume dropping them is an attack on them because it may, or may not, lead to best outcomes that they don't like.
A further point is the mention that dropping these assumptions make the problems intractable. There is a trad off between precision and accuracy. This is something we often see, research with better grounded assumptions can result in a more accurate picture but is so indeterminate that it can' be applied with sufficient certainty to be truly useful. However, this at least represents progress and does no harm. A more precise theory resulting from more tractable assumptions likely has the quality of being precisely wrong, it gives the illusion of usefulness due to its precision but because of its bad assumptions it will always tell you to do the wrong thing. Not only does this not progress the field but it actively causes real world damage if decision making is based on it. Better to make progress isolated to the ivory tower than to gain wide acceptance and do widespread damage.
*While not a scientific heuristic, the fact that people who hate the government are so invested in the non-scientific approach makes you wonder if the world in fact operates in the opposite fashion. It's not a good assumption, since the data isn't being measured we just don't know, but the fact they are so resistant to doing this makes you wonder if they've tried it and can't get the results they want.