Your wife will not be eligible for rent supplement until she has been assessed as in need of housing by the local authority, so that should be the first step. She can apply at her local authority office.
The other thing that she will need to keep in mind is that if this is really only a short-term break, that the HSE (and the Department when it comes to other payments such as OPFP) may not consider that you are actually separated and therefore take your income (and the value of the matrimonial house) into account when considering what payment your wife is entitled to. Have a read of the guidelines here on the means assessment process and in particular Section 3 "Assessing the Means of a Couple"
http://www.welfare.ie/EN/OperationalGuidelines/Pages/meansassess.aspx
If your wife is considered to be separated from you, then you still have to consider that any maintenance that she receives from you will be taken into account in the means test process. The way that this happens varies from payment to payment.
For rent allowance, the first 95.23 of maintenance is fully assessable, which means if you pay this much, that it will be deducted in full from any rent supplement which she will receive. There is then a 75 Euro disregard, which means that the next 75 Euro she receives will not effect her claim. Income after this is assessed at 75%, which means she will have her claim reduced by 75c for every 1 Euro that you pay in maintenance. I know this is a little convultuted, but if you let us know how much roughly you plan to pay in maintenance and which county you live in, then someone should be able to let you know how much this will affect her claim, or indeed if she is entitled to anything.
You can see that any significant amont of maintenance paid by you will seriously reduce her rent supplement payment.
As far as a welfare payment in her own right is concerned, your wife will have to wait 3 months before she can apply for One Parent Family Payment. She should apply for Jobseekers in the meantime to ensure that she has an application in the system (provided of course that she is looking for work). Once she has an application in the system she can apply to the Community Welfare Officer for Supplementary Welfare Allowance (SWA) until her main payment is sorted out.
The SWA rate is 186 Euro per week, plus 29.80 for each child. There is no disregard for maintenance payments, so any maintenance that she receives from you will be deducted from her payment.
To receive One Parent Family Payment (OPFP) she will have to show that she has sought maintenance from you. The rate of payment is slightly higher than the SWA rate at 188 per week, with the same 29.80 for each child. However the disregards are much more generous, the first 95.23 of maintenance (and the first 146.50 of earned income) are disregarded, and 50% of anything above this is deducted from her payment.
As far as freezing your mortgage is concerned, I would contact MABS and speak to a money adviser who can give you proper advice about how to manage your debt.