How long does it take from the moment that we send BTC to see our transaction in the ICO account dashboard? I've just sent my amount! 
ICO dashboard is updated every 30 minutes.
So it will show up in less than an hour. Keep the gas/tx fee high so that it doesn't get such up.
right, but sometimes it took an hour depends on the transaction fee you put on the transaction. On my case , it appears on my ICO dashboard for an hour due to to slow confirmation that time.