Can you add lace on a train?

I'd probably ask a seamstress as this will be who adds it. Most stores won't alter their original design after it's been purchased, but maybe a local seamstress would be able to tell you the answer.

It should for sure be do-able but you will likely need to find a seamstress who does not work at a bridal shop. They usually have limitations on the things they can alter. If you find the right seamstress they can usually make anything look good and will help guide you in the decision-making process of the changes.

It really depends on the dress itself. I imagine it depends on the seams and how it is structured.

Do you have a picture of the dress? Only a seamstress can tell you if it can be done or not.