There is not much background information when it comes to what your company is doing, but will try to provide some tips.
Assume you are working with customers and have some kind of ticketing system in place, so for:
1) I would be sending a very short survey with every single ticket just asking customers to rate the service they have received. Extremely well, Very well, Moderately well, Slightly well, Not at all well. This over time will give you very good indications on how the team is doing and how changes you apply impact customer satisfaction. There are many other important metrics when it comes to support but this one is key.
2) Again, no idea on what your product is about but i.e. in our case we monitor plenty of backend metrics. Successful/failed servers, successful/failed applications launched and so on for most important operations on the network.
3) Don't think that there is really a metric for that. You want to measure impact of new stuff you do in already present metrics and correlate success. Letting people innovate is more a culture related thing and how your organization embraces success/failure (as innovating implies failing many times and people will not innovate if they think they job is at stake)
In any case, for any metric to have any impact, it must be OBVIOUS to every one working for you. Make it AS EASY AS POSSIBLE for everyone to check metrics, try to make this a habit for key people in your organization if not everyone. We send a daily heart-bit to everyone with most important metrics and we have a grafana dashboard for each team, accessible to everyone with all historical metrics where people can thoroughly examine trends and see impact of everything we do.
Building a metric based culture, even if difficult is worth every drop of sweat it takes.