1. What does creating any third party rights in respect of deceased father properties mean?
You do not have to create third party rights when you have rights on your own, however in the maintenance case, you may create a charge over the property so that the karta does not alienate the same .
2. What does it is the responsibility of the KARTA to maintain and take care of the members of the HUF because the entire property of HUF is under his custody and maintenance mean or entail? As of today the KARTA is just taking care of property and not the people if he has to do anything for them what is he expected to do?
You have to ask the account details of the income derived from the HUF property and ask for your share in the profits or ask him to give amount for your maintenance as he is bond to do it, failing which you can approach court with an application for maintenance on the same basis.
3. I need clarity if I can actually claim maintenance out of my share in properties from your Karta of the HUF depending on the life style in spite of the fact I am educated, was earning well etc.? I have contradictory views here. Can I claim maintenance, without approaching the civil courts as that route would take a long time?
You can refer to the answer given above in Q-2.
4. I have documents proving that the HUF started by my father exist. How much and how regularly should I be getting my share from the earning of the HUF properties?
It depends, you can ask for monthly maintenance too.
5. How difficult or easy it is to separate myself from this recorded HUF ? Do you think I can or should do this before asking for entire property partition or after that?
If you want to exit out of the HUF, you can apply for partition by a suit in the civil court
6. I read online under Hindu law of inheritance that daughters also have a share in the mother's property. What does that mean?
The daughters do have a share in the mother' intestate property, what is your doubt about it?
7. In my case it would be exactly 15 years in Oct 2016 15 since my father has expired so can the WILL still be probated?
You can very well file a partition suit which is not barred by any limitation if the probate is barred by limitation.
8. Since you are unmarried, you have to right to stay in your demised father's property for your lifetime till you die or marry. What if the HUF house is now in my mother’s name legally now? Can I still stay there and ask for things/ facilities like car etc that they have deprived me of?
If your mother's property has been merged as HUF property then the rules of HUF will apply or if not you are entitled to a share in all her intestate properties as a legal heir to her, whether you are married or not.
9. So far my mother and brother claim ins pite of asking several times that there is no WILL of my dad. In case I file a legal suit and then they show up a WILL which they have hidden so far, what can I do then?