What is the inyan of actually touching\carrying the mita during a levaya? Is one allowed to visit the kever of a Tzadik within the first year?
There is a mitzvah of carrying the mita during a levaya, for this is actively taking part in the burial.
This principle emerges from Berachos 17b, where we find that those carrying the mita are exempt from Keriyas Shema.
See also Bach, Yoreh De’ah 361, citing from Roke’ach, who writes that the custom of all Jewry is to carry out the mes, taking turns in performing the mitzvah.
Concerning the great virtue of the mitzvah, see also Kesav Sofer, Yoreh De’ah 175; Maharitz Chiyus 6; Duda’ei Hasadeh25. Insome places the custom is that great and respectable people carry the mes; see Nitei Gavriel, Aveilus p. 319.
I do not know of any special significance of the first year with regard to visiting kivrei tzaddikim.