The number of free questions a user should be required to answer would depend on the desired quality of answers and the usage of the site. Answering questions is a time investment and your users have finite time.
High quality answers typically take much longer to produce than low quality answers. If it is fine for free questions to have low quality answers then you can set a large number and vice versa. You also need to balance this with the time constraints that your users have. The number of questions a user can answer would depend on the amount of time they are willing to devote to your site. If you set limits that they can't achieve users will not participate on your site.
You can determine the average time it takes to produce an answer of the desired quality and probably look at session times to figure out these numbers.
I've been doing some research myself on rewards in virtual economies. Given your approach, I think there are some additional factors you may need to take into account.
Volunteering as tax
If your site has paid questions then many of your users will join to earn money exclusively. Requiring users to answer free questions before they can earn money on your site makes a usually voluntary action mandatory. Volunteering doesn't function in this way so users who want to earn money will interpret this as a tax on their time. Users will be dissatisfied with this and probably rebel. Some will leave, some may stay and voice their displeasure consistently. Most will begrudge answering free questions and the quality of their answers will be poor.
Erosion of altruism
The presence of free questions on your site implies that altruism may play a role in motivating your users. There are at least two reasons I think altruism can become eroded as a motivation on your site.
Image motivation means that we care about what others think of our intentions. Monetary incentives for prosocial activities have been found to work best in private.
The results of the laboratory experiment, illustrated also in a field study, support our hypothesis
that monetary incentives depend crucially on visibility: monetary incentives are more effective
in facilitating private, rather than public, prosocial activity
Performing these types of activities in public makes us vulnerable to onlookers questioning our true motivations. Are we doing good because we want to do good acts or because there is an upside? Users who answer free questions for altruistic reasons will worry that others view their intent as qualifying to earn money thereby reducing their motivation.
Additionally, financial rewards can permanently erase internal motivations.
Results indicate that (a) when money was used as an external reward, intrinsic motivation tended to decrease; whereas (b) when verbal reinforcement and positive feedback were used, intrinsic motivation tended to increase.
Users who have joined for altruistic reasons will be motivated to answer free questions. But, since answering free questions will allow them to earn money they will soon become less motivated by their initial altruism. Over time they will view answering questions as transactions and will be predominantly motivated by the financial incentive.