One day internationals are defined by the ICC as occurring between senior teams representing their country in ICC sanctioned matches. Under 19s cricket is clearly a very different standard and it would obviously make no sense to roll these games into career ODI statistics.
Likewise, T20I cricket must be senior teams representing their country, so IPL, BBL or even CLT20 does not count as T20I nor would any U19 games.
The term 'First class' cricket refers to long form, non-limited overs cricket played at the highest standard within a country. This means for instance Ranji Trophy matches in India, English country cricket 4 day games and Sheffield Shield games in Australia. Club cricket does not count as 'First Class' because it is not the highest standard domestic cricket. Essentially First Class cricket it the closest thing to Test Cricket, but typically is played over 4 rather than 5 days however 3 day matches can still qualify as first class cricket. See here for more detail.
The term 'List A' refers to domestic limited overs cricket, again played at the highest standard within that country. This includes various competitions in which the games are between 40-60 overs. See the wiki page for more info on what does and doesn't count as List A.
Domestic T20 cricket includes games played in leagues such as the IPL, BBL etc. As far as I know the ICC has not yet codified exactly what constitutes official T20 records, i.e. presumably a Sydney park cricket T20 comp does not count, but where is the line drawn? In practice cricket stats sites use only stats from the premier T20 league played in a given country. But again, these career stats are different to a players T20I stats.
To see this in action, have a look at the stats as recorded by cricketarchive.com (which is officially endorsed by the ICC) for Mike Hussey. You can see clearly that the domestic records for First Class cricket are separated from Test cricket, ODIs are separated from List A and T20I is separated from T20 domestic. In addition, his under 19 records are again listed separately from his senior cricket records.
The only controversy in this area is the 6 day 'super test' and 3 one day matches played in the early 2000s between Australia and a World team. Some people don't believe these matches should count towards career stats for Test matches and ODIs respectively however the ICC has declared that they do count. The World Series rebel games played in the 70s also are not counted in career stats of players because these were not sanctioned by the MCC (the body that ran cricket before the ICC existed).