If I add a delay for a few seconds, and then will the value given by millis () be equal just before and after delay?
Or to put it in other words, does the internal clock of Arduino stop during delay?
No, I cannot check it (no Arduino at hand), but afaik the clock always keeps counting, so if you perform a delay of 1 second, and diff millis() just before and after, it will be (slightly over) 1,000 (ms).
Note however, that the use of delay is discouraged and it is better to check millis() to perform the next state/command instead of a delay which prevents doing other tasks meanwhile. But the (internal) timer always keeps running.
Only during some deep sleep mode it might stop, but than a delay will not work neither. And when interrupts are disbled (see remark of Chris Stratton below).
Delay uses millis. It just waits examining millis until a certain time has passed.
Millis is driven by an interrupt. The only thing that stops it working is another interrupt (you can only have one interrupt executing at once), which is why you can't use delay in an interrupt.