1. In Shakespeare's writings, and for his times, 'mistress' was a woman loved and courted by a man.
2. The sonnet is a parody. Most poets rave about the girl they love, using superlatives when they compare her as being 'fairer than any rose", "hair of pure spun gold', 'eyes of the most brilliant... etc. If any girl really was that beautiful, then her beauty would be rare indeed, and such a girl would be rare. But in fact, these comparisons are poetic, and belie ( do not give a true representation of) her looks, and just how quite ordinary looking the girl might be.
Shakespeare does not attempt to use hyperbole about his girlfriend. In fact, he refers to her flaws, and that she indeed is not perfection. Yet he considers that in her, he has found a girl just as rare as those girls depicted by other poets - poets who wax lyrical and exaggerate till their girlfriend is painted as the perfection of all that is beautiful.
'love' in the third last line refers to the girl, not his feeling for her.
And yet, by heaven, I think my love as (is as )rare
As any she : 'she' here does not mean 'she =the girl I love'.
'As any she' =as any other girl
belied with false compare.: misrepresented by the false hyperboles, the exaggerated statements (that poets usually use in describing the love of their life)
3. damasked : note the words 'damasked, red and white' - damasked means decorated as with a variegated pattern (and in this instance, the variegated pattern is red and white).