The limitations are not related to the distance. However, if to get the longer distance you’re using a lower data rate (a longer time on air for the same number of bytes) then you’re obviously using more of the limits for each transmission.
Yes, on the public (free) The Things Network is applies to all users (per node). However, it’s not yet enforced, so any problems you’re seeing are not related to this policy. It could be that the gateway is busy though, but then you’d get an error saying “no gateways available for downlink”.
When enforced in the future, it will likely be per 24 hours. So it might very well be some sliding window, unrelated to a calendar day. If you use all the limits right now, then you cannot use anything for the next 24 hours, regardless in which time zone you’re operating.
Again, your problems are not related to the Fair Access Policy. But you’re definitely exceeding the limits. A downlink every 10 minutes is really, really, not nice. That’s 144 per 24 hours, during which the gateway that’s transmitting those cannot listen to any uplinks. The same applies to confirmed uplinks. So, you’ll need to rethink your use case.