What will be the reaction of Jewish people if the Ark of the Covenant is discovered today? Will it bring joy?
this obviously may vary from person to person, but i think most jews will be happy about it.
What would they do with it?
no one would be permitted to touch it, since that requires purity from "tumah" (impurity), which occurs when even being under the same roof as the dead, or even a dead limb, and is transferred by touch. and to purify from this, a special concoction including dust of a burnt red heifer, is required.
Will it bring back the old practices of the Tabernacle System?
this issue of lack of purity will also prevent anyone from entering the temple area, (and a rebuilt temple is needed to perform services) let alone perform services, which would make the utensils tameh as well, which is a transgression.
there is a concept called "tumah hudcha betzibbur", which means we ignore impurity if most, or all of the jews, are impure. im not sure exactly how it will apply.
from a article titled "Why Don't We Rebuild The Holy Temple?"
Rabbi Moshe Shternbuch, in his work, Moadim Uzmanim,(14) in his treatment of the subject, ends off by saying:
And according to what we have explained at length above, there are innumerable reasons why we do not rebuild the Holy Temple or the Altar, nor bring sacrifices today. Nor does the repossession of the Land of Israel change that Law at all. We are unable, and therefore exempt according to the Law, without a doubt, for many reasons, until the Messiah arrives...G-d forbid that anyone should reconsider or doubt this...And I only discussed these matters out of interest in the subject, due to love of the Holy Temple and the Holy Service.
May the Holy One, Blessed is He, pour upon us a spirit of purity from high above, and may we be found worthy of having G-d's Holy Manifestation in our midst when G-d returns the Service to His Sanctuary speedily, and with our own eyes may we merit seeing everything straightened out.
in addition even with purity, your average joe of the street would also not be allowed to touch it. see here, also see here