If the parent company wholly owns the subsidiary, could the employees of the subsidiary work on products of the parent company? Are special contracts needed?
Yes. Think of it; plenty of multinational companies have development centers all over the world.
Yes, and this is much too situation-specific and complex for Answers.OnStartups. In general, you would need some kind of framework around a) transfer of intellectual property rights, including patents, trademarks etc, and b) transfer of money, including taxation issues.
Most young startups sidestep these issues by not creating a subsidiary .. just hire a service provider (an 'outsourcing' company) in the remote destination, and buy a service from them.