Can you count time as a parameter?
No. The configuration is what potentially changes over time. So the equations of motion are a function from time, into the set of configurations. Time is the domain and the set of configurations is the codomain and the equations of motion is the function from the domain to the codomain.
Can't you always increase the number of parameters (even if it has no effect) and still determine the configuration of the system?
They would not be independent.
In the case of a pendulum, most texts say it has only one degree of freedom,
That's the planar pendulum, confined to rotate in a plane, like a grandfather clock.
In the case of projectile motion, the projectile has 3 degrees of freedom, right?
Yes. At each point in time you have to specify three coordinates to specify the configuration at that time. More if it is extended and can have orientation, even more if it is not rigid.
Keep in mind that a degree of freedom is about the space of possible configurations. It isn't about any one particular equation of motion.
Can someone give me a rigorous definition of the degrees of freedom and explain how this definition addresses the questions above?
The only place it seems you stumbled is about independence. You should be able to freely adjust any of the coordinates in the degree of freedom within some little bit and have a different configuration.
But this is also a false generality. For $N$ particles the degrees of freedom is $3N$ and sometimes you can pretend there are fewer by pretending that some constraint is exact when it is not actually exact. For instance a real pendulum can and does elongate a little bit and the place it pivots can wiggle a little bit and so forth. The one degree of freedom is really about ignoring the other degrees of freedom.
So just have $3N$ and then start eliminating ones you don't care about whose dynamics hardly change in an important way. And just don't over eliminate, you should retain enough to describe your system. In the case of the pendulum when you know the end point and you assume the rigidity and one part fixed, then you know the whole thing.
What can you really gain by pretending to have more generality than is really there?
The answer depends on interpreting the question. But my answer would be three, because it can (1) rotate in one spot, it can (2) move along the rod, and it can (3) move around the rod.
An insect typically has six segmented legs, four wings, and various other moving body parts. All those involve several degrees of freedom, so if you include them in your count, your answer will be a lot. Presumably, you're supposed to ignore all those degrees of freedom, and consider the insect as a more rigid object.
Now, think of the insect as a rigid body, but one that is able to move around. It could maybe go up on its hind legs, for example, which would be another degree of freedom that I didn't list above. But I tend to think that you're supposed to just imagine the insect walking around in the most boring way possible. That involves just walking anywhere on the rod (two degrees of freedom because the rod's surface is two dimensional), or turning around in one spot (one degree of freedom because it just takes one angle to say which way the insect is pointing). That adds up to three.
Best Answer
The minimum number of independent generalized coordinates is given by the number of degrees of freedom. Nothing prevents you from using more coordinates (dependent) if you so desire.