At least in the United States, time transitions from 1:59:59 AM EST to 3:00:00 AM EDT. I don't know if other countries have different rules, but it would be surprising to me if they did.
The changeover time is just as much an arbitrary political decision as all the other aspects of DST change. So yes, you can find a variety of changeover times. You can find them in the "AT" column in the rules files.
Changeover times obviously vary, but I'd be surprised if anyone set up their rule such that the time change happened immediately after the changeover time rather than at the changeover time (eg, at exactly one second after 3:59:59 it was 4:00:00, but one second plus epsilon after 3:59:59 it was 5:00:00). In practical terms, it wouldn't actually make any difference, but I would find it surprising if anyone's rules were written that way. - Adam