You don't choose a subset of your original 99 (100-1) variables.
Each of the principal components are linear combinations of all 99 predictor variables (x-variables, IVs, ...). If you use the first 40 principal components, each of them is a function of all 99 original predictor-variables. (At least with ordinary PCA - there are sparse/regularized versions such as the SPCA of Zou, Hastie and Tibshirani that will yield components based on fewer variables.)
Consider the simple case of two positively correlated variables, which for simplicity we will assume are equally variable. Then the first principal component will be a (fractional) multiple of the sum of both variates and the second will be a (fractional) multiple of the difference of the two variates; if the two are not equally variable, the first principal component will weight the more-variable one more heavily, but it will still involve both.
So you start with your 99 x-variables, from which you compute your 40 principal components by applying the corresponding weights on each of the original variables. [NB in my discussion I assume $y$ and the $X$'s are already centered.]
You then use your 40 new variables as if they were predictors in their own right, just as you would with any multiple regression problem. (In practice, there's more efficient ways of getting the estimates, but let's leave the computational aspects aside and just deal with a basic idea)
In respect of your second question, it's not clear what you mean by "reversing of the PCA".
Your PCs are linear combinations of the original variates. Let's say your original variates are in $X$, and you compute $Z=XW$ (where $X$ is $n\times 99$ and $W$ is the $99\times 40$ matrix which contains the principal component weights for the $40$ components you're using), then you estimate $\hat{y}=Z\hat{\beta}_\text{PC}$ via regression.
Then you can write $\hat{y}=Z\hat{\beta}_\text{PC}=XW\hat{\beta}_\text{PC}=X\hat{\beta}^*$ say (where $\hat{\beta}^*=W\hat{\beta}_\text{PC}$, obviously), so you can write it as a function of the original predictors; I don't know if that's what you meant by 'reversing', but it's a meaningful way to look at the original relationship between $y$ and $X$. It's not the same as the coefficients you get by estimating a regression on the original X's of course -- it's regularized by doing the PCA; even though you'd get coefficients for each of your original X's this way, they only have the d.f. of the number of components you fitted.
Also see Wikipedia on principal component regression.
Best Answer
I think mathematically, what you are attempting to do may be feasible (answer to question 1). As far as interpretation of betas (question 2), PCA is already extremely challenging to interpret as is in the best of cases. Usually, most of the explanatory power is concentrated in the first Principal Component. When studying the composition of the Betas on the variables of that first Principal Component, you often observe results that are counterintuitive and cryptic. And, extending interpretation to the second and third Components is most often as baffling.
PCA is best used for two reasons: 1) streamline a large number of independent variables into three Principal Components; and 2) resolve issues of multicollinearity associated with a very large number of independent variables. However, PCA has major drawbacks. It is so opaque (opposite of transparent). It renders clear interpretation of the results from very challenging to impossible.
An alternative to PCA is simply to streamline your model. Use a clearly defined and straigthforward dependent variable. Explore tens of independent variables if you wish, but select judiciously the best 6 or 7 ones that make the most sense supported by social sciences and economic theory. Usually, this generates a simple, explanatory model that is easy to interpret and present to various audiences. You can add more variables, but watch out for model overfitting. The latter is actually a mathematical trap that PCA models can readily fall into (overfitting the data, meanwhile being very poor predictors because they fit to the noise within the data).