Saints do nothing on their own power, but they are humans created by God who served Him well on earth and continue to serve Him in heaven. The Catechism reminds us that there is a communion of the Church between Heaven and Earth. The following quotes found in the Catechism explains the intercession of the saints:
CCC 956 The intercession of the saints. "Being more closely united to Christ, those who dwell in heaven fix the whole Church more firmly in holiness…(T)hey do not cease to intercede with the Father for us, as the proffer the merits which they acquired on earth through the one mediator between God and men, Christ Jesus…So by their fraternal concern is our weakness greatly helped."
Do not weep, for I shall be more useful to you after my death and I shall help you then more effectively than during my life (St. Dominic, dying, to his brothers.)
I want to spend my heaven in doing good on earth. (St. Therese of Lisieux)
So as I understand this, they not only pray, but they offer the “merits” that they earned during their life through Jesus Christ.
I don’t know if this is explained in the Catechism, but “patron saints” typically relate to the works they accomplished on earth. For example, St. Dominic Savio died when he was just a teenager and he lived an exemplerary life, so he is a patron saint of school boys because he showed how a school boy can follow Jesus well. Sometimes the connection is rather odd, for example St. Stephen, who was stoned to deat, is the patron saint of stonemasons and bricklayers. I think of “patron saints” as “specializing” in certain areas of Christian life.
Of course you can ask any saint to pray for you regardless of their “specialty”. As Jesus has perfected them, their prayers will be more perfect than what yours or mine might be. God is generous with His friends, and He shares His glory with the saints as they in turn glorify God first in their lives on earth and now in heaven.