You're mistaking race and ethnicity.
With the greatest degree of respect, you do not understand the area well enough to tutor others about it. You have no understanding of genetics or anthropology; you would not make such egregious errors if you did.
I will illustrate your errors in layman's terms so that others who seek the truth may find it, despite your almost certain desperation to ignore good science in favor of your brand of stoned schoolboy reasoning!
There is no coherent, fixed definition of race.
True. However, it does not means that races are not useful taxonomic categories. There is no coherent, fixed definition of the breed of dog known as the German Shepard. Assuming that you apply your rules consistently, you must thus agree that since we cannot absolutely define what a German Shepherd is, German Shepherds do not exist.
This is clearly a dumb thing to think, just as is it a lil' bit dumb of you to think that races are not useful taxonomic categories. You can see how foolish you claim is now.
racial categories are based on appearance, not genetic background (hence "socially constructed").
Anthropologists categorize people into a group by assigning them to the group of which they hold the most genomic information from. Certain jet-black skinned populations in India are categorized as Caucasoid, since they have more Caucasoid information in their genome than either of the other popular divisions. They do not, as you seem to believe, look at them and say "you have black skin: you must be African or a dark skinned Asian"
Racial categories are not based on skin tone, they are based on the fact we can group populations who have endured similar evolutionary pressures by DNA markers. Physical anthropologists can categorize the race of an individual to a high degree of accuracy without any DNA testing by looking at their bones.
So, you see, the structure of your skeleton and your DNA profile tell the same story. Everything you have said apart from there being no fixed definition of race is wrong, and you misunderstand and misapply that information as an amateur.