He can't marry Sansa to Harry without revealing who she really is, and she's still married to Tyrion so that marriage must first be annulled. This was why Littlefinger helped the Tyrells frame Tyrion for Joffrey's murder, to have him executed and free Sansa for future use as a bride. Unfortunately for him, Jaime and Varys screwed that up.
And I don't see why Sansa trusts him or would be willing to go along with his schemes. She should know that Littlefinger was the one who betrayed Ned in the first place. It won't surprise me if she ends up just sticking a dagger in him when she realizes he's going to kill Sweetrobin (her cousin) to advance Harry.
He undoubtedly knows that Cersei's children are illegitimate, but he keeps that information to himself. Perhaps he wants to have something to hold over Joffrey later. When Jon Arryn finds out, Littlefinger persuades Lysa to kill him. When Ned Stark finds out, Littlefinger betrays him and lets him die. But then Littlefinger himself has Joffrey assassinated later.
Stannis was the first one to figure out that the kids were illegitimate, and he told Jon Arryn so they began investigating Robert's bastards to realize their suspicions. Varys and Littlefinger knew something was up and spied on them, figuring out the secret for themselves.
LF had Arryn killed for two reasons: 1) So he could use it and Lyssa to frame the Lannisters, then alert whoever the next Hand would be and hopefully start up a War. He'd then weigh his options and choose whoever he thought was the likely winner and demand a reward (hopefully a promotion to High Noble) so he could marry Lyssa and become Regent of the Vale. Everything else was just him adapting to random things as best he could and trying to benefit from them. He's just been really lucky more than anything else.
2) He had to keep anyone from de-legitimizing Joffrey in a controlled peaceful way, because that would make Stannis the next in line and Stannis would likely outlaw all the stuff LF makes his money from. That is, if Stannis didn't outright just kick him off the Council.