You should start by claiming Jobseeker's Benefit (JSB) - which is based on your PRSI contributions. The SW office will tell you if you have enough contributions.
The advantage of JSB is that it's not totally means-tested, so you should qualify for full rate (currently €188) for yourself, and you may get an increase for your wife and child.
If you don't qualify for JSB, you'll be given an application for Jobseeker's Allowance (JSA). Your wife's earnings will be taken into account, but there are some disregards, so you may still qualify for some payment.
If your wife is working 19 hrs or more per week, she might qualify for FIS. Your JSB or JSA will be assessed as income when calculating entitlement to FIS. If your wife qualifies for FIS, you will not receive an increase for her on JSA or JSB.
Regarding your older child: One of the qualifying rules for receiving a child payment is that the child must be "ordinarily resident" with the person claiming. There is a paragraph in the guidelines (link below) which deals with children of separated parents, where the child isn't living with the person making the claim:
Where
the parent with whom the child is residing is not in receipt of a Social Welfare payment (other than Child Benefit, Guardian's Payment (Non-Contributory) or Supplementary Welfare Allowance)
AND the other parent is contributing substantially towards the maintenance of the child
The child may be deemed to be normally resident with the other parent, and would not be considered normally resident in any other household. " Contributing substantially" in this context is taken to mean at least the current equivalent of the Increase for a Qualified Child.
What this means is that if the other parent of your 14 yr old is not claiming a SW payment (other than one of those listed), and you are contributing at least the equivalent of the child increase (currently €29.80 pw) for the 14 yr old, the child will be considered to be "ordinarily resident" with you, so you can claim for them.
Link:
http://www.welfare.ie/en/Pages/Increase-for-a-Qualified-Child.aspx