In S7, the timer has two parts: the time base (as you noted) and the timer preset value, which is a BCD number from 0 to 999. A timer decrements by 1 at each interval as indicted by the time base.
So your timer with a time base of 10ms and a preset of 1 will decrement to 0 (expire) after 10ms. The timer updates at the interval chosen (10ms in your case).
Your green online indication should remain green in single scan or in continuous scan unless your program faults or was not successfully downloaded initially. Try a master reset and re-download your program to see if that solves your issue.
Also, I am using the PLCSim v.5.4+SP8. Earlier versions could have had a different behavior when in single scan.