I've been wondering about #1 as well, so I e-mailed the representative from my host country (Australia) for clarification. Apparently the source of the monthly allowance varies between countries. In most cases, the funding comes from the NSF-counterpart of that country. However, in certain countries, it may be up to the host to find funding for you during your stay. This might be through their own research funds or the university's research office. I'm not sure what the situation for the GROW program is in France, but it might be worth contacting them for clarification if you're unsure.
I'm still trying to figure out how this will affect my application. Can anyone who went to Australia for the GROW program comment on their experience? Did the host university have a funding mechanism in place for GROW scholars? Did your host lab need to provide funding?