First, thermodynamic equilibrium is not exactly analogous to mechanical equilibrium, in which a system is truly in a mechanical constancy. At any given point within in the system at a given time, there will be small fluctuations from true equilibrium. However, such fluctuations are vanishingly small for large macroscopic systems in what we describe as equilibrium.
Second, a state of thermodynamic equilibrium is a highly relative property. It is a valid approximation under the proper temporal and spatial ranges. For example, a new copper pipe may be well approximated as in an equilibrium state with it's environment on the order of years, but will corrode on the over of decades. A puddle on a cloudy day may be approximated as in thermodynamic equilibrium, but over several (even cloudy) days, will evaporate. In terms of spatial equilibrium, at a given time, a certain gas may approximate equilibrium on small, localized volumes, but may exhibit global nonequilibrium. For a rough example of this, take the atmosphere directly around your home on a mild day. At a given time of day, the pressure may remain at a stable value, however, if we were to record the pressure for your country at the same time, the assumption of equilibrium would be invalid.
All of this is to say that, when we define a state of thermodynamic equilibrium as a state of a system where certain state variables remain static in space and time, we mean, over appropriate spatial and temporal scales, we can approximate these macroscopic variables as stationary.
As far as your question about slow expansion/compression of a gas, this is an entirely different assumption. In thermodynamics, a system's state doesn't change unless we alter its state externally (i.e., add heat or perform work). Now, thermodynamics is strictly concerned with a system in equilibrium. If the time it takes the system to relax to an equilibrium state once an external interaction is applied (call this $\tau_{r}$) is much larger than the time is takes to apply the interaction ($\tau_{i}$), than the system will be out of equilibrium significantly longer than the change that caused it does. However, if $\tau_{i}>> \tau_r$, than the system can be approximated to instantly return to a state of equilibrium as the change is applied. 273
This approximation is known as a quasi-static process. For an example of a quasi-static process, imagine a cylindrical container of gas with a pump handle at the top. The gas starts with an equilibrium pressure $P$. The gas also has a specific wave speed $v$, at which waves will propagate through it. If the speed at which you compress the gas at is much slower than the wave speed, the pressure in the gas will roughly stabilize at each increment of the compression. This is a good approximation of a quasi-static process.
As for Question 1, the concept of a quasi-static process is integral. The assumption that a P-V curve is parameterized by time is incorrect, as the system's state variables must be static over time and space, unless acted on externally. What the P-V curve on a P-V diagram shows is the state evolution of the system as a quasi-static process is enacted on the system. That process could be implemented over any arbitrary time interval, so long as the assumption of quasi-staticity.
As for Question 2, I'm a physicist, so internal combustion engines are not my expertise, but I believe it all comes down to idealizations again. It's not the motion of the pistons or other engine parts that we consider in equilibrium, but rather its thermodynamic state variables. As you mention, this is of course not reality. What we typically use thermodynamics for in relation to engines are idealized engines, in order to quantify upper bounds on operating limits. We often assume quasi-static cycles on P-V diagrams to model the engine cycle and quantify properties such as heat exchange and work performed. The idealized engine is the Carnot engine, from which a multitude of information is gleaned, such as maximum engine efficiency between two temperatures, the thermodynamic temperature scale, etc. 568