Part 1
Small eigenvalues are not included in the definition of stiffness for ODE (initial value problem) systems. There is no satisfying definition of stiffness that I know of, but the best definitions I've come across are:
If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a step length which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval. (Lambert, J. D. (1992), Numerical Methods for Ordinary Differential Systems, New York: Wiley.)
An IVP [initial value problem] is stiff in some interval $[0,b]$ if the step size needed to maintain stability of the forward Euler method is much smaller than the step size required to represent the solution accurately. (Ascher, U. M. and Petzold, L. P. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia:SIAM.)
Stiff equations are equations where certain implicit methods, in particular BDF, perform better, usually tremendously better, than explicit ones. (C. F. Curtiss & J. O. Hirschfelder (1952): Integration of stiff equations. PNAS, vol. 38, p. 235-243)
The Wikipedia article on stiff equations goes on to attribute the following "statements" to Lambert:
- A linear constant coefficient system is stiff if all of its eigenvalues have negative real part and the stiffness ratio is large.
- Stiffness occurs when stability requirements, rather than those of accuracy, constrain the step length. [Note that this "observation" is essentially the definition from Ascher and Petzold.]
- Stiffness occurs when some components of the solution decay much more rapidly than others.
Each of these observations has counterexamples (though admittedly I couldn't produce one off the top of my head).
Part 2
Probably the best example I could come up with would be integrating any sort of large combustion reaction system in chemical kinetics under conditions that result in ignition. The system of equations will be stiff until ignition, and then it will no longer be stiff because the system has passed an initial transient. The ratio of largest to smallest eigenvalue should not be large except around the ignition event, though such systems tend to confound stiff integrators unless you set exceedingly strict integration tolerances.
The book by Hairer and Wanner also gives several other examples in its first section (Part IV, section 1) that illustrate many other examples of stiff equations. (Wanner, G., Hairer, E., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (2002), Springer.)
Finally, it's worth pointing out the observation of C. W. Gear:
Although it is common to talk about "stiff differential equations," an equation per se is not stiff, a particular initial value problem for that equation may be stiff, in some regions, but the sizes of these regions depend on the initial values and the error tolerance. (C. W. Gear (1982): Automatic detection and treatment of oscillatory and/or stiff ordinary differential equations. In: Numerical integration of differential equations, Lecture notes in Math., Vol. 968, p. 190-206.)