Involutions, RSK and Young Tableaux

For question 2, there was a paper published in 2013, Standard Young tableaux and colored Motzkin paths. http://www.sciencedirect.com/science/article/pii/S009731651300099X. They used "colored Motzkin paths" to characterize standard Young tableaux, and I think "colored Motzkin paths" is something like non-crossing partition. Roughly, the level step is a fixed-point in the involution and the pairs of up and down steps are the cycle in the involution. The most important things in this paper is that they found the relation between "bounded height SYT" and some restrictions of "colored Motzkin path".