Is gas cost of external calls included in gas cost of the main call?

When I call external methods of other contracts from my contract, is the gas fee of these calls included in the gas fee displayed by MetaMask?

My transaction reverts (execution reverted) for an unknown reason. May it be that it is because the price of inter-contract calls is not included into the gas price displayed by MetaMask, and for this reason my calling contract has not enough funds to pay for external calls?

If this trouble exists, how to calculate the full gas cost in advance, to be sure the entire transaction succeeds?