Placing ice into a glass of liquid: 
One may place ice into a glass of wine or water during the summer in order to cool down [the drink] into which [the ice] melts on its own. One need not worry [that doing so transgresses any prohibition].
Placing ice into a glass of liquid to melt:
It is permitted to place ice into liquid. [Although to crush it inside the liquid, or into the liquid is forbidden, as explained above in Halacha 1B].
May one pour liquid onto the ice in ones cup?
It is forbidden to pour the liquid onto the ice being that doing so causes it to melt.
May one pour some water onto an ice tray to help break the ice?
Yes, as this is similar to breaking a piece of ice which is allowed due to that it only melts a minute amount as well as that the excess liquid goes to waste.
May one pour water onto salt/sugar and other dissolvable items?
Yes, as they are hard and do not dissolve through the pouring and thus are not similar to the case of urinating on ice. Seemingly one may even pour hot water from a Keli Sheiny onto these substances as the amount that it will melt is only slight. Nevertheless regarding instant coffee since it dissolves immediately with the placing of water into it, it is best to prepare coffee essence with from before Shabbos and thus avoid any suspicion of prohibition. Nevertheless if one did not do so he may do it on Shabbos itself.
May one mix the ice cube in his drink using a spoon or the like? 
No, as by doing so one is actively causing the ice to melt which is similar to washing ones hands with icy water of which one is to be stringent against doing so.
May one place a dissolvable item such as sugar and instant coffee into liquid and then mix it with a spoon so it dissolves? 
May one place soap into a cup of liquid and have it dissolve and then use that to wash ones hands?
 Piskeiy Teshuvos 320/5
 And it is thus similar to urinating on ice which is to be avoided. See Ketzos Hashulchan 127 Footnote 14
 Az Nidbaru 6/5
 Ketzos Hashulchan 127 footnote 14
 Shabbos Kehalacha Vol. 1 p. 266
 So is evident from all Poskim which have never written of such a prohibition and some [SSH”K] have even allowed placing instant coffee into a Keli Sheiyni. So is implied also from Shabbos Kihalacha ibid.
 So is implied from Ketzos Hashulchan 127 footnote 2. See also SSH”K 10/2 and footnote there.
 Ketzos Hashulchan 127 Footnote 2
 Seemingly doing so should be forbidden just like it is forbidden to crush ice with ones hands even into a liquid. Nevertheless the custom is to permit doing so, due to the fact it is not really the mixing of the spoon which dissolves the sugar but really the liquid, and the mixing only quickens the job. [ibid]