Rollergirl1 - Your husband was, I assume, baptised, confessed,communioned and confirmed, he IS catholic.
There is no get out bar death or conversion.
Get them Baptised by all means BUT be prepared to jump through hoops when applying to the school of your choice.The parish priest will probably want to see proof of baptism.Your MIL's priest should ask for dispensation to baptise your children as it is an out of parish child(ren).Dispensation is fun to get.You can send your child to catholic school even if they are not catholic.
Your MIL probably see this as a big deal so feels your mother should be there to see it.It is the first of the sacraments in the catholic faith so YES it is a big deal to your MIL.
Your godparents need to be catholic, this is quite important as you are promising to bring up the children in faith,the priest will be quite strict on that.You should also attend a course beforehand.This can vary from 1 session to 4 months of weekly sessions which also the priest can want to see the godparents attending some.
And yes you can have a proxy Godparent, I stood in for my cousin when her godmother was unable to attend, this had to be run past the priest first-he allowed this being as the Godmother was in hospital at the time having given birth 3 months early.