Answered your questions as follows:
1) A judge in Hyderabad has recently dismissed our case on the grounds that none of our lawyers were present at the court hearing. This case was pertinent to a property dispute between our family members. I was just wondering if in this situation I could register my house with my dad's will who is deceased.
Ans: In such case before it is reopened or appeal/revision is preferred you have every right to act according to your wish and get transfer the house in your name as per Will of your father.
2) Furthermore, I was wondering if any of my family members could potentially reopen this case and what its status would be once the said case is reopened?
Ans: Yes, by filing a miscellaneous application by other party it can be reopened but only after serving a copy upon you.
3) Due to the unforeseen circumstances of my father's death, my mom is currently left as a widow. So I was wondering if my mother can write a will pertaining to properties on my father's name and if the said will would be viable?
Ans: No, your mom cannot write Will pertaining to properties standing in the name of your father. For more issues please call me.