Mother died leaving unregistered will wants to transfer the property

You first need to get it reflected in official record, only then you will be considered the owner of share of your mother's share and will be entitled to lease it out. Even after that there is scope of challenge by your siblings which shall be contested only and you will have to prove the Will in your favour if at all it is challenged.

Yes You can lease out pending Transfer as well, in the capacity of landlord.
Your father cannot claim share in the property left by your father.
Your siblings can unless you prove the will and get it implemented

Since the share of your mother has not been transfered in your name on the basis of unregistered will , you can,t lease it out till the property is transferred in your name. Your father or other Lrs of your mother can challange the will. If you succeed in proving the will , only then you can get the share of your mother. other wise an unregistered will is also a valib document if it is proved or not contested by other Lrs of your mother.

You have to get probate of Will from Civil Court, without probate of Will property mentioned in Will can not be transferred in your name. You have to prove Will by gill my probate petition and thereafter the properties will be transferred into your name