Lets see if we can make some sense of this confusion.
Start from what does a unit mean.
A unit length, for example, could be
1 meter
1 inch
1 foot
1 yard
1 kilometer
1 mile
1 stadium
1 parasangue
etc.
Units need definition and conversion factors from system to system, particularly if one wants to build a bridge or plan a road.
Ratio's of quantities where the units are eliminated are universal, whether you are talking of meters or parasangues ( an ancient persian equivalent of kilometer length)
Take the perimeter of a circle and divide it by its diameter. Whatever units you may have used to inscribe the circle, the ratio is pi, whether a kilometer diameter or an inch diameter.
Given the radius of a circle, whether it is small, in inches, or huge, in kilometers, one can find the perimeter in the appropriate units by multiplying by 2*pi.
This and similar quantities simplify the work not only for geometers, in map making, engineers and architects, but all scientists.
The same is true for units of weight ( don't let me make a long list of them). The ratio allows easy communication and calculations whether for tons or pounds.
etc.
There is no answer to this. When you are taught to use dimensional analysis at school the teacher invariably selects an easy example (it's almost always the pendulum) to keep things simple. In the real world there is no guarantee that you have a dimensionless constant.
It's actually quite rare to use dimensional analysis to derive equations in the real world. The sorts of simple systems that are amenable to dimensional analysis are usually already well known. However it's very, very useful to use dimensional analysis to check that an equation you derive is dimensionally consistent.
For example suppose you're working through a differential equation for some quantity, and after covering many sheets of paper with scribbles you end up with a final equation. It's very easy to make a minor mistake along the way, so the first thing you check is that your final equation is dimensionally consistent, i.e. the dimensions of the left and right sides are the same. If they aren't that means you've made a mistake somewhere. I routinely do this in my answers to questions on this site!
Best Answer
Here are two examples of where dimensional estimates fail:
Divergent expressions
I have a laser pointer pointed at a wall directly facing me, 1 meter away. I turn the laser pointer by 90 degrees over the course of 1 second. What is the average speed of the laser spot during the turn? The dimensional analysis argument is 1meter/1second, which would be roughly right if I didn't have a singular function. Similar examples are resonant phenomena, where you have a lot more response than you would predict from dimensional analysis.
Exponential growth:
Suppose I have 20 particles in a box of mass M, in volume V, with total energy E. When will they all return to within 10% of their starting positions/velocities? If you ignore the dimensionless 19, then you will get the recurrence time for one particle, which is shorter by an enormous amount.