Which knots would you say are the most important to know and why?
Nominating specific knots is tricky for a variety of reasons: there is often more than one that will do the job; the type and size of cord, and the details of the application, are important factors (e.g. a hitch, such as constrictor hitch, that works on a round pole may not work so well on a square pole); the basic knot may not be suitable but a variation or extension of it might be (such as when adding a lock or doubling the collar on a bowline); and personal preference/allegiance also plays a part.
To try to make a simple list or algorithm is not possible, there are too many ifs and buts, it is too conditional. At least, nobody has done it yet, far as I know.
But I will say it is important to know at least one reliable, well tested knot from each of the basic categories of knots, and to know the limitations of a knot.
To take an obvious example, an Alpine Butterfly loop is a very good fixed non-directional midline loop. It is the first one I teach anybody for this category. Reasons for this choice include that is is very well tested in the field, can be tied easily in the bight, can be inspected easily (very important from a safety angle), doesn’t jam easily (meaning it can be undone without much trouble, when required), and also works well as a bend. But it would not be my first choice for a fixed end line loop, though it will work well enough as one.
For a fixed end line loop, the choice depends on factors like: whether you need ring loading capacity or not, what safety level is required, etc.
Some of these factors are more generic to all knots, like safety level, others are more specific to a limited set of knots, like ring-loading capacity.
And so on…
This is for strictly functional knots. Requiring a decorative element adds another variable.
Which knot would you say you use most often and is most fundamental for you? For example, if you were asked to tie a knot for a purpose you don't already know a knot for, which would be the first knot you would go to to try?
I don’t believe this question can be answered in the abstract. It depends on the concrete details of the application. I would suggest that there are already enough basic knots to cover almost any application you can think of, either with a single knot, or a combination of knots.
In effect, what you are asking us with these questions is: does the knotting world have a coherent system of logical principles for describing and classifying knots, and the practical choice of knots. And the answer, far as I can tell, is no. It is still very much an empirically based enterprise, with guiding rules of thumb.
Maybe you can fix that for us. ;D
All the best.