It is certianly falsifiable. If a coin can be shown statistically to favor one side over the other consistantly in a statisically signifigant data set. And the results can be duplicated consistantly then the claim can be shown as false. The size of the dataset is going to depend on the margin of error. The greater the margin of error (or wider of the statistical distribution) the larger the dataset will need to be.
Does falsifiability require the process of falsification be finished in finite time?
No, but the in order to be falsifiable you must have the ability to demonstrate a false condition. In order to determine is someone has lived for ever we need to measure the length of their entire life. In order to measure the entire length of someones life it would have to end. If it the life ends then it is not in a false condition.
If a coin will land on its head 9 out of 10 times And i can demonstrate this repeatedly and independant of biasing conditions, then i can demonstrate a false. If it is biased at a much lower rate say by 1 per 1 million. It would require a much greater dataset but i can still show the bais confirming a false condition.