If you want to use the verb engage, it sounds better if you say, "She engages in charity work." or "She engages in charitable activities." Charity work usually refers to working through an organization. Charitable activities can done alone, with others or through an organization.

This is an indirect answer to your second question, and a cultural note. In English-speaking countries, if you wanted to give someone the very highest praise, you could call them "a peacemaker." This is higher praise than most heads of state deserve. It would be reserved for people whose careers seemed to be devoted mostly to promoting peace--not just negotiating a treaty. It would be reserved for the kind of people who have been awarded the Nobel peace prize (which Obama, in fact, has). It is praise so high that most US presidents would not deserve it. Because of politics, very few presidents would be seen as "peacemakers" by everyone, and although I used the Nobel peace prize as a measuring stick, that prize always controversial, too. I personally would call Jimmy Carter "a peacemaker," although very likely some people would not agree. "Peacemaker" has the plain meaning of "someone who makes peace. However, this particular word carries that particular tone because it appears in the Bible, spoken by Jesus: "Blessed are the peacemakers: for they shall be called the children of God." It's part of a long speech by Jesus called "The Sermon on the Mount." It's in a section called "the beatitudes" because it is a long list of people he says are "blessed." Like Shakespeare, this passage is so famous that parts of it here and there have entered the language. Normally, we say "blessed" as one syllable, as if it were spelled "blest," but in this passage it's customary to read it as two, "BLESS-sed." It's referred to as "Matthew 5:3-12." Blessed are the poor in spirit: for theirs is the kingdom of heaven. Blessed are they that mourn: for they shall be comforted. Blessed are the meek: for they shall inherit the earth. ... Blessed are the peacemakers: for they shall be called the children of God.

I would say "she does charity work." In the case of Obama, "Obama tried to promote world peace" or "Obama tried to advance world peace."