I am a Navy Chief. I work at Navy Nuclear Power Training Command. Please take my advice and use it to make a sound decision. You'll be able to get through the schooling just fine. It's what happens after your training that you should be concerned about in regards to you family. Your family and your marriage will be severely strained when you graduate from Nuke school and go to the fleet. Nuke school will take you 18 months or so and then you'll go out to sea. That means that you're oldest child will be starting school shortly after you hit the fleet. You'll be out to sea and you'll miss that time (a normal sea tour lasts about 4 years). And your second child will be in school before you get done with your first sea tour. It's just not worth it if you're just trying to find a job. Joining the military is a lifestyle change...not a job change. There are other ways to provide for your family. If you have an extremely strong family, you'll make it through. And the Navy will offer tons of opportunity, but it is a long term commitment (6 years minimum) and it's not for everybody. But, to answer your other questions....
You can still succeed.
It will be more difficult because you have a family. Nuke school can take up a lot of time, so you'll have to be able to prioritize between your family and the Navy.
I know several mothers who are doing just fine in nuke school. I also know some moms who are not doing so well. It all depends on the individual.
Hope this helps. Feel free to PM me if you have any other questions. I'm sure that some other people on this forum will have some good advice too. Good luck!