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there are only 2 cavalry regiments in the army; 2nd at ft polk and 3rd at ft carson. there is a "regimental" designation system that has a battalion and then the regiment ie 4 bn 64 armor "regiment" (4/64 ar) unlike WW2 the battalions do not fight as regiments. tsome battalionsd will be in the same brigade, others will not. so with the exception of the 2 armored cavalry regiments (2nd and 3rd) nobody else is actually a regiment. the 1st cavalry division is an armored division not a cavalry division. each division has a cavalry squadron (equivalent to a tank or infantry battalion) for reconaissance. 3/7 cav is most likely the recon squadron of the 3rd infantry dividion. 1/7 cav, if it exists, is in another division.
i'm sure thaey honor the linrage of the 7th cav from the ia drang campaign and earlier campaigns. when i was in germany in 1978 there was a 7th cav squadron assinged to 8th ID or 1 AD not sure).