(1) If you want to minimize the 1-norm or the \infty-norm of some variables, then it is possible to write an equivalent problem that has a linear objective (with some additional continuous variables and linear constraints). However, if the norm expression appears in constraints, then the linearization may be more complicated, involving integer-valued variables.
Your descriptions of the linearizations seem to have the right idea, if you are looking to minimize the norms. But to get more help, you should post the AMPL formulations that you are using. You don’t need to give the whole model, just the part that shows how you linearized the norm.
(2) If your equivalent problem has a linear objective function, and all of the constraint expressions are linear, then you should be using a linear solver. If these solvers are successful, you can expect them to find optimal objective values that are the same to many decimal places. AMPL works with many linear solvers, including CPLEX, Gurobi, Xpress, COPT, MOSEK, HiGHS, SCIP, and CBC.
If you have any expressions in your model that are not linear, then to get more help, you should post some examples of how those expressions appear when they are written in AMPL. (Again, you don’t need to give the whole model.) Note that local nonlinear solvers like MINOS and Knitro are not guaranteed to converge to solutions that have the same objective value.
(3) Since you are the person who has the best knowledge of your application, you have the main responsibility for verifying that the model and data are correct. However, you can use this forum to post specific questions about optimization using AMPL.