No, Shadow doesn't have a romantic relationship with anyone.
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No despite Rouge flirting with him.
SEGA officialy rejected storyline of game Sonic the hedgehog 2006.Also they accepted that Rouge likes Knuckles.
According to the Sonic 06 Official Guide Rouge has a crush on both Shadow and Knuckles, but this is yet to be confirmed by official sources.
Rouge The Bat. She wish she can been shadow and trying her best to get shadow to love and taking her a long time and hard to get shadow to like you for a start and love is much harder so its rouge the bat who loves shadow
yes shadow dos't know it but i can tell he has a soft spot for rouge and soner or la8er he will tell her the truth As far as we know, Shadow hasn't really taken an intrest in Rouge, but Rouge has seemed to taken intrest in Knuckles. (Well, not really, but we know that Rouge flirts with him a lot to make him irritated)