I think the questions are very broad (too broad), so this answer will be correspondingly (too) shallow.
It's 'community wiki' in case you want to edit (add to) it.
Do you believe that Buddhism has shaped society today. If so how?
Yes.
It does so by providing an alternative. If your question is like, "Do you believe 'that door over there' (Buddhism) has shaped 'prison' (society) today?", then Buddhism might be about freedom.
There's an article titled The Taste of Freedom which includes,
"Just as in the great ocean there is but one taste — the taste of salt — so in this Doctrine and Discipline (dhammavinaya) there is but one taste — the taste of freedom": with these words the Buddha vouches for the emancipating quality of His doctrine.
How has Buddhism influenced your life?
I don't think that's on-topic on this site, because that isn't common/shared experience.
Where in Buddhism does it say that you have to live your life in peace?
There's a general concept called non-violence or Ahimsa.
More specifically there are rules of ethics which are called e.g. five precepts: which include "peaceful" rules like "don't kill" and "don't steal" and "don't lie".
Do you believe that a pursuit of peace is necessary in today's society?
I think that people (in society) find stress unpleasant, and that too much stress (or greed) can lead to war and destruction of society. Even if there weren't inter-personal or international problems, there are still personal problems: for example your friends and family (and you) getting sick and old and dying.
So, yes, some alternative to or relief from stress, some peace, is necessary to society.
Various members of society (i.e. people) learn about Buddhism when they feel motivated to.
As a Buddhist, how does a pursuit of peace begin? (How do you pursue it, what does a "pursuit" of peace entail and when does your pursuit end?)
I think it begins with stress. When you experience stress then you pursue peace.
A way to pursue it by following the Buddhist "noble eightfold way" a.k.a. "three-fold training".
The pursuit ends when it's successful.
This is summarized in the Four Noble Truths:
- Stress exists
- Stress has a cause
- There is a way to end stress
- The way to end stress is the "noble eightfold way"
There is more than one school of Buddhism though. In this answer, with the "four noble truths", I'm trying to summarize the earliest school, but see also e.g. the last paragraph of this answer which suggests that the pursuit ends when one is, also, liberated from the stress of (or perhaps peaceful even within the constraints of) Buddhism. This answer suggests there some real difference in the end-stage of different Buddhist schools (but that's probably Too Much Information i.e. more detail than you were looking for).
Do you believe that all people should follow a life of peace?
Yes and no. I don't expect lions to stop hunting. I think people people might be happier and hurt each other less if they were more Buddhist.
As a Buddhist, do you think that Buddhism has the idea of changing the world?
Yes and no. Buddhism is mostly or originally about changing yourself.
Buddhists also have an effect on each other:
Ven. Ananda said to the Blessed One, "This is half of the holy life, lord: admirable friendship, admirable companionship, admirable camaraderie."[1]
"Don't say that, Ananda. Don't say that. Admirable friendship, admirable companionship, admirable camaraderie is actually the whole of the holy life. When a monk has admirable people as friends, companions, & comrades, he can be expected to develop & pursue the noble eightfold path.
Buddhists may also have an effect on society at large (see for example Did the Buddha try to effect social change?) and includes advice for laypeople who continue to live within society.
Furthermore, Mahayana Buddhism includes the concept of not only liberating yourself but also "changing the word" (e.g. liberating "all other sentient beings").
If Buddhism values finding peace within yourself, how is each person finding peace among themselves going to help change the world?
For the better, presumably.
One way is that the purpose of "virtue" is "lack of remorse". In order to find peace, people may find they need to become more effective (e.g. so they can fulfill their social duties).
The Dalai Lama says that happiness comes from making other people happy.
