0. Start to start is a type of relationship/restraint/dependency, not a type of activity.
1. There are many scenarios where SS relationships are useful to model workflow in summary, where the detail necessary to use FS relationships is not justified. Then SS+lag is applied where the lag represents a certain volume of predecessor work that must be done before the successor can start. The lag may be only a millisecond, but the precedence exists nonetheless.
1a. A negative lag (i.e. a "lead") with a SS relationship seems almost exclusively for date-manipulation. I would question the underlying logic basis.
2. Whether one or both activities are on the Critical Path depends on upstream (predecessor) and downstream (successor) logic for both activities. There is nothing special about the SS relationship in the CPM/PDM calculations.
3. If a) the SS relationship actually drives the successor and b) the successor's downstream logic places it on the critical path, then most modern CPM software will also assign the predecessor to the critical path. The whole bar is colored red, though in fact only the start should be considered "critical." Saving Changes...
Start to Start is a very common relationship. The most common types I can think of are where Group B requires some input from Group A to begin their own tasks, prior to the completion of Group A's deliverable. For example:
Group A is responsible for the design of a product. Group B is responsible to plan the production of the product. Group B cannot start planning production, until Group A has started the design, or there is nothing to base the production plan on.
This is often on the critical path, because the two were scheduled in parallel rather than series in order to save time. Otherwise, Group A could complete their work prior to Group B beginning. Saving Changes...