As a runner, making a run is considered an action with a cost of one click (page 15). During a run, can you play and event card like Tinkering, or do you have to wait for your run action to finish first before you do another action?

TBossAZ said:

No, you have to wait until you finish your run. You need to use Tinkering before you make a run (before you know what you're going to encounter if there is unrezzed ice) to benefit from its effect.

Thanks for the clarification, much appreciated.

To clarify the rule itself: because you spend a click to make a run, you cannot do any other actions that would require spending a click until the current action (in this case, the run) is resolved.

Whilst on the subject of Tinkering I note that it is applicable to a piece of ice. I assume you would have to play additional Tinkering cards before running against multiple ice protecting a server and place the Tinkering cards alongside the ice for the duration of the turn until discarding.

You don't HAVE to.

Technically Tinkering goes in the heap after it is played, but it doesn't hurt anything to put it next to the ice as a reminder.