Single parent tax credit can be claimed by any parent who has responsibility for their child and is not living with the child's other parent. In my experience in this area, Revenue do not seek evidence in regards to over night access or written access agreements.
Only one home can be nominated as the Family Home and the transfer of this property is not subject to tax, be that Stamp Duty or CGT. The proceeds of the sale of the 'non-Family Home' (for want of a better description) will be subject to tax in the usual way.
In regards to maintenance payments: payments made for your spouses maintenance can be set against your taxable income, but is taxable as income in your spouses hands. Child maintenance cannot be set against your taxable income. If your spouse is not working it can make sense to pay a higher level of spousal maintenance and a lower level of child maintance.
There are different rules that apply to tax in regards to the year of separation; but I don't recall them clearly off the top of my head. I doubt you'll make any money on it though
In regards to TRS; to the best of my knowledge ( and I'm willing to be corrected on this) you can only claim TRS on one property.
The best piece of financial advice I can give you in regards to separation (with apologies to all lawyers) is to keep the lawyers out of it if you can. They don't do much that you can't do for yourselves provided you can communicate and can often make matters worse by use of legal language that puts backs up.
Family Mediation is free and is very helpful. There are offices in Tallaght and in Dublin City Centre. A phone call to the main office will give details of an office nearer to you if necessary.