Speculatively, GMan is 'in control' of the timeline of the Half Life series - certainly in the opening speech he gives in Half Life 2 he gives the impression it's him pulling all the strings. There are more references in his final speech of that game.
Given this, it's probably for the same reason he 'extracted' Gordon Freeman (or led him to safety, depends how you read his words from that speech I referenced earlier). I'd say it's because Alyx is somehow needed to save the world.
Perhaps it could be said that GMan is somehow a 'protector' of the world, and that Alyx and Gordon are chosen as heroes for specific abilities they possess. Alyx certainly seems to have an affinity with technology (see 'Dog' character from the series, and her proficiency with the gravity gun).
I think this is really as deep as one could go without resorting to making stuff up - it's already highly speculative and based on pretty thin strands of evidence.
Black Mesa is a fan made total conversion mod of HL2, recreating the original Half Life. A team of volunteer artists, coders and level designers have painstakingly recreated the beloved HL single player experience, adding vast amounts of new details such as more scenery objects, more detailed landscapes, and richer textures. They even recorded new dialog to freshen that up too.
Half Life: Source was a port by Valve of HL to the Source engine. There was no new content added and the textures werent updated, so the only differences were better lighting, shading, and physics. It looked a bit better, but played the same, and was really only ported as an experiment to test Source's modding capabilities.
In short, HL:S was a better way to experience HL if you'd never played it by the time HL2 arrived. BM is a chance at reliving all the magic again, but in a very new and handcrafted way.
Best Answer
No. The Black Mesa mod is a high-def remake of Half Life 1. That's it. It uses the Source engine to do so; it's not a Half Life 2 mod, persay, as much as a total conversion. Half Life 2 was used as the starting point, but is not at all required to play Black Mesa.
What the mod page DOES say is:
Play for FREE! You need not purchase anything!, and classify it as a Half Life 2 mod. This conflicts, and the correct answer is the bullet point. Most likely, it was classified as a Half Life 2 mod because the database doesn't allow it to be blank. A better place to look is the official site.HL2 also uses the Source engine, but is in no way required to play Black Mesa.