Hello Aureliebg!
There are a lot of posting that address this question, so you can read through them on this forum. I am an American who went through the process, with my Dutch husband. We were married before coming to Ireland which is a big plus! We also have advanced degrees, so our situation is somewhat similar.
I know you are confused, but you must understand before you make the jump to Ireland that getting a residence card for your husband is going to be tricky. Since you are French, you will be able to stay in Ireland without any issues. However, your husband will have to apply for an Irish residence card using the EU1 form. This process may take up to 9 to 12 months before you get an answer! I waited over 7 months.
During that time, your husband is "technically" permitted to remain in Ireland. However, if you are leaving the country while waiting before he has been granted residence, he may be hassled at customs. During this time, your husband will not be permitted to work.
You should be aware that if your husband does not obtain a French residence card prior to coming here, there is a good chance that his application for residence will be DENIED. Yes, even though you are married with children. Irish government mandates that a spouse of an EU citizen must have residence in another EU country prior to coming to Ireland. This decision was just ruled on in the Irish supreme court. I think as long as he has a French residence card, he should be fine.
However, if you cannot get a French residence card for some reason, bear in mind that because you were married for several years prior to coming to Ireland, there is a possibility that your husband
may not be denied. If you fortunate, some of us (myself included) were granted a Stamp 4 Visa for 2 years...which gives your husband permission to reside and work in Ireland. In essence, a Stamp 4 is similar to a residence card but for a shorter period of time.
If he can get a work permit through a place of employment, that's great. Not every employer is willing to provide that, though. However, trying to get a work permit through your EU spouse has been suspended as well, so he cannot get a work permit through you.
IMPORTANT: When you apply with the EU-1 form, even though they don't ask for it, give any other documentation you feel will support your case. Also, definitely submit copy of private insurance cards...I hear that again and again that you should send in those. If you have the money, you may want to consider hiring legal represenation.
Hope this makes sense. If not, PM me. I wish you and your family all the best through this pain in the arse process!!!
and Archigabe, I don't think the Irish administration is the last remain of last century, 
for better and for worse.