To which Saints may a Catholic pray, and when can this begin?

The basic question here is about Catholic “praying” to saints. It is not exactly a prayer, but it is like asking saints or even Mary to pray to God on our behalf. Catholic only ask for their “intercession” not pray to them.

Regarding sub-questions:

Which saints are qualified to receive prayers?:
Only those are venerated as saints, who are designated as Saints by the Catholic Church, intercessory request are made to them.

What is people prayed to John Paul II or Mother Theresa today, but they receive the designation of Saints at a later time?:
People pray to God that through the intercession of these “would be saint” some miracles may be performed.

Some instances in Bible verse:

Rev 5:8
and when he had taken the scroll, the four living creatures and the twenty-four elders threw themselves to the ground before the Lamb. Each of them had a harp and golden bowls full of incense (which are the prayers of the saints).

These saints in heaven are offering prayers to God under the form of golden bowls full of incense, which are the prayers of saints. Meaning that saints offer our prayers to God, and they hear our petitions and present them to God by interceding for us.

The intercessory requests to Virgin Mary and Saints are akin to everyday life instances. When I am in dire need of prayers for any of my intention, I would ask my friends and well wishers also to pray for me. So those in heaven have even greater confidence and devotion to God than anyone on earth. Also God hear the prayers of righteous.

Asking people to pray for you no way violates the mediator ship of Christ. Christ is unique mediator incomparable with human mediator ship as He is both God and Man and is the only bridge between us and God. This is also supported by Paul in four verses immediately preceding 1Timothy 2:5.

1Timothy 2:1-5
First of all, then, I urge that requests, prayers, intercessions, and thanks be offered on behalf of all people, even for kings and all who are in authority, that we may lead a peaceful and quiet life in all godliness and dignity. Such prayer for all is good and welcomed before God our Savior, since he wants all people to be saved and to come to a knowledge of the truth. For there is one God and one intermediary between God and humanity, Christ Jesus, himself human.

It is similar to saying of a son humbly at his mother’s grave, “Mom, please pray to Jesus for me. I am having a real problem right now.” We in our daily prayers, pray for our children as they being young, would not exactly know how to pray and does not mean we are taking Christ place for our children while directing our prayers to God for them.

Paul directly asks others to pray for him:

Rom. 15:30-32-32
Now I urge you, brothers and sisters, through our Lord Jesus Christ and through the love of the Spirit, to join fervently with me in prayer to God on my behalf. Pray that I may be rescued from those who are disobedient in Judea and that my ministry in Jerusalem may be acceptable to the saints, so that by God’s will I may come to you with joy and be refreshed in your company.

And finally Jesus himself required us to pray for others, and not only for those who asked us to do so:

Math. 5:44 But I say to you, love your enemy and pray for those who persecute you

To which Saints may a Catholic pray, and when can this begin?

Actually a catholic can privately pray to anyone whom he/she thinks can intercede on their behalf. In fact Catholics do not see any difference in asking you to pray for me and asking my dead grandma (if I believe she is in heaven or purgatory) to pray for me.

Is it restricted to only those who are designated as Saints by the Catholic Church?

Public prayers are restricted only to canonized persons^{see note 1}. But private prayers/veneration is not. In-fact the actual canonization procedure can start if and only if for 5 years after the death of a person, the lay faithful have prayed to them privately and continuously. But such prayers can not be done publicly in a Church or Chapel. More specifically their statues and pictures cannot be placed in an alter of a church/chapel. They cannot be represented with halo around their head.

Is it the designation of sainthood by the current Pope on earth that allows someone in heaven to begin hearing prayers?

No it does not. The designation of sainthood or canonization only attests that the universal church can pray to them because this person is in heaven and has the power to intercede before God for us. The main purpose of Canonization is to show people Good example. This is revealed by infallible papal proclamation. Canonization is helpful only for people on this earth to recognize a saint, it does not give special powers to people in heaven. Private devotions are not restricted by this.

What prevents any Christian from hearing the prayers of people on earth? Would it have something to do with the time when they get out of Purgatory?

On the contrary, a catholic can even ask for the holy souls in the purgatory to intercede. Only thing is that souls in the purgatory cannot pray for themselves, but they can very well pray for us.

Points to note:

1. Persons who are beatified can be publicly venerated only in the concerned persons diocese. For public veneration outside of his own diocese the permission of the Bishop or the Bishop council of the country is required. Such restrictions do not apply for one who is canonized. For example Bl. Mother Theresa is permitted to be venerated all thru India by the Catholic Bishops council of India, even though she belongs to the Archdiocese of Calcutta.
2. There are some saints whose devotion is explicitly suppressed because faithful have been having devotion to them, but their life is not a good example or there is little to no proof for their existence. For example as late as 1173 Pope Alexander III, reprimanded certain bishops for having permitted veneration of a man who was far from being a saint. Also the names of St. Valentine and St. Philomina are removed from universal church canon, because of lack of evidence for them to have existed. But their veneration is permitted in local churches where it is traditionally held.