Minimum Rests...
No. Whenever you're back in your home port it goes back to the 12hrs or 15hrs (if the shift was over 8hrs).
Also, we (and major airlines) don't do 'split shifts', but some airlines (regionals) can make you work in the morning, supply a hotel for you to rest near the airport, and then you work again in the afternoon! Now that's a short rest!
When you get an interview for a company, and post your queries, I'm sure someone will PM you with the exact information for your chosen company.