Multiple linear regression with lm() in R, why is the intersection dependent on the name of the "first" country I have a question about the function lm() used for multiple linear regression analysis.
Context: We have a dataset (that I cannot share) where $y$ is the proportion of women (in some context that I cannot share), then we have some independent variables $x_i$ (numerical values).
Hence I make the command line lm(y~x_1 + x_2 + … +x_n, data = dataset) and that works well. I get a point of intersection n, + more.
However, the dataset allows us to use country as independent variable. For each country at least five points for each independent variable exists. Our hypothesis is that we have a global trend, however er would like to allow for cultural differences in offset. Thus, we are aiming for the model:
y_{country} = a + ( \sum_i  b_i * x_{i,country}) 
    + \epsilon_{country}

Thus, we have the command line lm(y ~ country + x_1 + x_2 + … +x_n, data = dataset).
Problem: My problem is that is seems that the analysis is then dependent on the country that comes first in the alphabet i.e., Australia. In the example with Australia, I get a point of intersection a. But if I change the name of Australia to ex BB, then the output is dependent on Austria instead, and the value of the point of intersection a’ is different from a.
Questions: My questions are: why is the point of intersection dependent on the “first” country? Is there a way to change this? Preferably I would like the “base” intersection of the country specific analysis to be n, and then the (output) list gives the country specific difference from the base intersection (n-a).
I hope my question is clear, and I am sorry, but I cannot share my data with you.

All the other responders make valid points.  But it's also important to say that, regardless of the coding used, the differences between countries will remain the same.
The reason the output changes when the "first country" changes is that country is a factor.  A model with both a factor and an intercept is over parameterised.  So (at least) one model parameter can't be estimated.  Different software packages will choose to handle this in different ways.  R chooses not to estmate first factor level, essentially equating this term with the intercept.  Coefficients for other levels of the country factor then get interpreted as " the difference between country x and the first country".
The important thing to realise is that the estimates of "the differences between country x and country y" will reman the same, regardless of the parameterisation chosen.  This is true for all x and y.
