Yes it can, but you need to install a small ledger under it until it drys enough to hold it into place. Not sure how long it would remain as its not really used for this purpose. I am assuming your doing this on a finished wall of some sort?
I personally would however use pre-mixed mastic cement used for gluing tile to walls. Its designed for this and would most likely work much better then the out of tube adhesive.
If you have a wood wainscot below the Ledger may not be needed. I have alos used blue painters tape to secure small tile bits.
I agree with KMS on this one (yet again). Just use the blue painters tape to hold them in place while things dry. Preferably overnight.
There are a couple of fast grab const adhesives. I think one is from Locktite.
You can also accelerate cure of these by mashing the tile into it in place, then pull it off and waiting a few minutes before mashing it back in again, but I still use tape overnight to be sure
Also, forgot to mention - when you use the adhesive, make small circles on the back of the tile and don't just put the stuff along the edges. that way, when you mash it on the wall, it will create a bit of suction and help keep it in place.
Yep, dabs rather than beads let you even place it relative to the next one too
If you want something that is more permanent use polyurethane caulk or the white mastic that is used for setting wall tile. The polyurethane is most likely going to be your best choice and the mastic second choice. I would not recommend or use construction adhesive. As it gets older it gets hard and loses it flexibility so that if it is bumped it will break off since it becomes brittle as it ages.
I'm a big fan of the Acrylpro mastic. it is available in small containers of a qt up to 3.5 gallon. You can get the quantity you need easily