Equilibria: in the macroeconomic sense of aggregate equilibrium where all markets clear, markets are most likely never in any equilibrium but rather in constant flux between different equilibria, because the market clearing macroeconomic equilibrium always depends on real and also in short run nominal factors which constantly change. Hence it does not make sense to say whether the markets are currently in equilibrium or not. However, textbooks use the concept of such equilibrium because it is very useful way of describing states toward which the markets gravitate - you can imagine them as a screenshot of a continuously playing movie.
Completeness: this condition, in Arrow-Debreu sense, requires there to be very small or non-existent transaction cost and perfect information, this is almost definitely not satisfied in many cases and consequently you will see a lot of research that looks at what happens when there are transaction costs and information asymmetries in markets.
Arbitrage: empirically arbitrage opportunities are definitely present in financial markets but they are usually small (i.e. no million dollar bucks left laying on a side walk) and they tend to disappear really quickly for example many studies of covered interest parity in foreign exchange markets show that arbitrage opportunities usually don’t last more than few hours, sometimes even seconds and arbitrage opportunities lasting few days are already considered long (Mark, 2000). History also shows longer arbitrage windows but they usually come from times where information technology was not as sophisticated as now. So you could say that the condition of no arbitrage in real life holds most of the time approximately.
