About marriage legality according to Hindu - padmashali caste

Your Father's aunt's daughter will be an aunt to you. Her daughter will be sister in law to you. So you can marry her. But in some communities marrying the father's sister's daughter is not considered. But a girl can marry the father's sister's son. But in your case, she is your father's first cousin's daughter only. So there will not be any problem in marrying her.
As per the Hindu Marriage Act, 1955, your relation will come under "Sapinda relationship". As per the act marriages which will come under Sapinda are barred. if the bride is a person within five generations including himself on the groom's father's side and three generations including himself on the groom's mother's side, or vice versa, they will be "Sapindus" of each other.
But legally your marriage will come under the special Marriage Act 1954 and will be registered under this category only. Legally there will not be any problem. But you have to make a decision based on your family member's decision or you and the girl has to think and make a decision.
My elder son married my mother's sister's granddaughter and it will be registered under the special marriage act. There is no problem. Both of them are doing good and they have two daughters.

In some Hindu castes and communities there is a tradition of marrying in near cousins. If it is so then they are exempt from the application of Hindu Marriage Act, 1955 in this regard. Otherwise as per this act it is clearly mentioned that up to fifth generation in the line of ascent through the father (considering the present generation as first) and up to third generation through mothers side (considering the present generation as first) the relationship is considered as Sapinda relationship and marriage would not be allowed within that. So we have to trace the common ancestor and find out that.

In your case it is well outside the Sapinda relationship so you can go ahead with this match. One thing to be noted in this reference is that these laws are basically enacted to protect the interest of people and their legal claim in the ancestral property and another thing is that there are cases when people have married within the Sapinda relationship but did not have any legal problem as no one complained about them in this regard and there was no property dispute where such relationship could be exposed. So the legality of the matter comes in picture only when there is a dispute and someone challenges that marriage itself is illegal so benefits cannot be availed by the contenders. One going for close relation marriages should well understand this fact.

Another point to be noted is the law does not differentiate between the normal child versus adopted child and considers it same for its application in such cases.

There is no restriction on marrying in a family Gotra anywhere in the Hindu marriage act. In this case, two adults of the same gotra can marry, provided they are not in a Spinda or Prohibited relationship. But your relationship is not under it, you can marry your father's sister's daughter with the consent of your family.

As you said, that girl is the daughter of your father's mother's sister's daughter, so this relationship will not be considered prohibited so you can marry with her.
Marriage cannot occur in forbidden relationships. Brother-sister, uncle-niece, aunt and nephew, brothers' children, real brother and sister's children, maternal uncle and niece, grandfather or maternal grandfather's brother or sister, grandmother or grandmother's brother or sister, brother's wife Prohibited relationship Come under the purview of and in such a relationship if someone gets married then it will not be valid. Spinda relation cannot marry. That is, if a boy or girl wants to get married in the 5th generation of the father and the 3rd generation of the mother, then she cannot get married. This is called the Spinda relation. However there are exceptions to the Prohibited relation and the Spinda relation. If there is a custom of such marriages in a community then it can be married. Like in many areas there is a tradition of marrying a brother's wife. Similarly, the marriage of maternal uncle and niece is prevalent in many places in South India. These marriages are valid despite having a prohibited relationship.

Your father's sister is your aunt. So you have no problem in marrying her daughter as you have right to marry her. The said girl's mother is your father's mother's sister's daughter. This girl's mother is cousin sister to your father and however she is coming in the line of sister to your father and thereby that lady become aunt to you and her daughter is a girl you can relatively marry.
As for as relationship you can marry her. But nowadays a practice is prevailing that scientifically marriages within relations have some genital problems. So far I concerned the marriages withing right relationship is acceptable and valuable than the outside marriages as the incoming girl can know the customs and rituals of our family than the incoming girl. Many difference of opinions, disputes are coming only based on the rituals and follow ups as the same is insisted by mother in law to daughter in law. In many houses the gents do not care on the rituals and follow ups as the society is expecting this only from house ladies. So, the husbands newly married do not care about the rituals and followups and they do not care if the newly wedded girl do not follow them and even he got angry with his own parents who insists or expect the followups from the girl. If the girl is within the relationship, there are possibilities that she could follow them.

The relationship as stated by you, the girl is your distant sister and this relationship is not going to affect your matrimonial relationship provided you consider the following points -
1) You should closely examine her temperament such as her reliability, patience, the attitude of sacrifice , a good listener and above all she has the sense of undertaking the decision depending upon the situation.
2) Since the relationship of the couple deteriorates with the belief of loose talks of your close relatives and your known aquiantences, she should go in depth of the matter before it spoils the relationship.
3) Both of you must offer due respect to the relationship of either side so as to enjoy the lasting relationship of marriage.
4) Your fiancé must have the capability to adjust with the family members of yours especially with your parents taking enough care of their needs and in that way, both of you will enjoy the happy relationship.
5) It would be better for both of you to undergo for medical examination determining the genes effect, and other related disorders such as high/ low Blood Pressure, HIV test, gynaecological disorders obstructing her for the normal conception. Such tests prior to marriage, will help you in understanding the disorder and if needed, medical assistance can be undertaken.