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.

It makes sense that the intercept will change when you re-order a factor variable that is used an independent variable in a general linear model.  The best way to see this is to fit the first model, then fit the model with the factor variable re-ordered.  Then choose some values for country and for the other independent variables, and plug them in to both models, and you'll see that both models predict the same value for the dependent variable.
I don't know if it would satisfy your purposes, but an alternative approach would be to treat country as a random effect, and use a mixed effect model.  In this case the order of the levels of country won't matter.  (And this might be a better approach for your data set.)
library(lme4)

library(lmerTest)

model.1 = lmer(Y ~  IV + (1|Country), data = Data)

summary(model.1)

Edit, addition:

Preferrable 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).

The way to do this would be to set the intercept to zero.  (In R, you can do this by including a -1 on the right-hand side of the model formula.)  Usually this isn't a desirable approach. Instead, usually, you would want to use e.g. emmeans() to show you the differences in effect of the levels of e.g. country.
