Insignificant at the 10% level means that the 90%-confidence interval overlaps with zero. A significant difference at the 1% level means that the (larger!) 99% confidence intervals do not overlap. This should not be possible.
The F-test does not test the hypothesis whether two coefficients are of different size. This test tells you that large firms are not small firms. What you are probably looking for is the Wald test. Maybe that’s the problem?
Results from an ordered logit/probit regression are always unintuitive, but categorical explanatory variables are as meaningful as continuous ones. I'd even say that they are easier to interpret.
For a concrete example, you could look at Dobson, An Introduction to Generalizer Linear Models, 2002, 2nd ed., Chapter 8. In her "car preferences" example, the dependent variable is the importance of air conditioning and power steering (three levels: "no or little importance", "important", "very important") and the two explanatory variables are gender (male or female, coded as 1 and 0) and age (18-23, 24-40, >40, coded as age2440 = 1 or 0, and agegt40 = 1 or 0).
Fitting an ordered probit model you get (I've used R, MASS library, polr() function):
Coefficients:
male age2440 agegt40
-0.3467 0.6817 1.3288
Intercepts:
NoImp|Imp Imp|VeryImp
0.01844 0.97594
Then you can compute the probabilities for women (male = 0) over 40 (age2440 = 0, agegt40 = 1):
NoImp Imp VeryImp
0.095 0.267 0.638
and for men over 40 (male = 1):
NoImp Imp VeryImp
0.168 0.330 0.502
Their difference is the gender partial effect:
NoImp Imp VeryImp
-0.073 -0.063 0.136
I think that it's meaningful ;-)
Best Answer
With one ordinal variable with 4 levels, you should only have 3 threshold coefficients. You are essentially dealing with an unobserved continuous outcome which has been divided into four observable buckets (the levels). The thresholds are the boundaries between the buckets, so there should be only 3. The variable $\log_{10} (x)$ and the slope parameter $b_4$ determine how you move from one bucket to the next as the unobserved variable changes.
Here's an example that might give you some intuition.