Fitting Lasso model repeatedly on same train dataset
In principle, that shouldn't happen.  If you rerun Lasso on the same data set, with the same hyperparameters, after dropping attributes whose coefficient was set to zero in your first Lasso run, you should get the same result.  If you see something different, something has gone wrong.

You'll need to investigate deeper to see if you can figure out why that is happening.  Perhaps there is a bug in your code, or perhaps you are not providing the right settings to your implementation of Lasso, or perhaps the optimization algorithm is not finding the exact optimum solution, or perhaps something else has gone wrong.  It's hard to say with the level of detail you have provided.

If you're using different hyperparameters the second time you run Lasso, anything could happen.  Don't do that; that doesn't make sense.