If I have a continuous dependent variable ($y$) and independent variables including an ordinal variable ($X_1$). How do I fit a linear model in
R? Are there papers about this type of models?
Generally there is lot of literature on ordinal variables as the dependent and little on using them as predictors. In statistical practice they are usually either assumed to be continous or categorical. You can check whether a linear model with the predictor as a continous variable looks like a good fit, by checking the residuals.
They are sometimes also coded cumulatively. An example would be for a ordinal variable x1 with the levels 1,2 and 3 to have a dummy binary variable d1 for x1>1 and a dummy binary variable d2 for x1>2. Then the coefficient for d1 is the effect you get when you increase your ordinal for 2 to 3 and the coefficient for d2 is the effect you get when you ordinal from 2 to 3.
This makes interpretation often more easily, but is equivalent to using it as a categorical variable for practical purposes.
Gelman even suggests that one might use the ordinal predictor both as a categorical factor (for the main effects) and as continous variable (for interactions) to increase the flexibility of the models.
My personal strategy is usually to look whether treating them as continous makes sense and results in a reasonable model and only use them as categorical if necessary.