No worries, thank you for the additional information.
Given that there are no orders in place yet, the rules of a Rice & Asplund won't apply to you. They only apply when changing existing orders, of which there are none in your case.
The first step toward parenting orders is to organise a family dispute resolution conference with the mother to try and reach an agreement about care arrangements for the child.
This is a mandatory step before court proceedings can be pursued, but it's helpful to go into family dispute resolution with some idea about what the court would focus on if it were to decide the matter for you.
First, the court will only uphold one person's rights in parenting matters, and that is the child's rights to know, spend time and communicate with both parents and other people relevant to their care on a regular basis, insofar as the child's best interests can be met. If those rights are not currently being upheld, the court will do its best to make orders that ensure the child is given the opportunity to have a meaningful relationship with both parents, provided its in their best interests to do so.
Second, orders made by the court in parenting matters are only in the best interests of the child, and the governing legislation that guides how the court decides on the best interests of the child is section 60CC of the Family Law Act 1975. Have a read of that - it'll give you an idea about what the court would consider in your case, but the primary consideration is the benefit to the child of having a meaningful relationship with both parents, and the second is the need to protect children from harm. There are about 12 other considerations as well.
Third, your ex is incorrect about her proclamation that she has sole custody of your daughter. The term custody is not used in Australia. What is used instead is the term 'parental responsibility', which refers to the responsibility of the parents to make major long-term decisions about the child's care, welfare and development, which includes living and care arrangements, education, name changes, religion and medical. All parents, including you, share equally in parental responsibility unless removed by order of the court (and that doesn't happen very often).
Where shared parental responsibility exists (as it currently does for you), the court will consider whether an equal time arrangement is in the child's best interests, or if not, whether substantial and significant time with be non-resident parent is in the child's best interests. Substantial and significant time is a combination of weekdays, weekends, holidays and special occasions.
There are some weaknesses in your case, such as having little contact with your daughter for five years, not paying child support and having what I assume is some difficulty communicating with your ex, but if there's no domestic violence issues or other threats to the child, the court would likely order that the child spend time with you on a regular basis so that their right to know you can be upheld.
Be prepared though - the court would probably look at a gradual reintroduction of time between you and your daughter, and would probably order counselling to help rebuild the relationship, so be ready to welcome those opportunities. Also, court is a slow process, so hopefully agreement can be reached via family dispute resolution before you have to pursue court.
Summarily, do I think you can get the current arrangements changes? Yes.
