1) No, the 18-gon is not _compass_and_straightedge_ constructible from scratch. However it is constructible with other tools. I won't extend this topic any further, construct the 18-gon is equivallent to angle trisection. A n-gon is constructible iff n = 2^k product of distinct Fermat primes. (that is 3, 5, 17, 257, 65537, and no others are known)
2) IMHO, an angle is "given" iff it is so mentionned in the problem assignement ;-) All others are deduced, either obviously like when two angles of a triangle are given, then the third is deduced. Or most often after a more or less complicated proof, which is precisely the aim of the problem ! Also note that giving an angle extends compass and straight edge possibilities. The given angle may be not constructible ! The construction is now no more "from scratch" but "from a given figure". From a _given_ 40 deg angle in the asignement, we can easily construct the 18-gon.
About angles which are "guessed", this corresponds to what you wrote: "Assume that the unknown angle measures xx degrees. Calculate..." Guessing the result often gives a hint to solving. However just "calculate... " doesn't generally suffice to solve Sujeet's puzzles, as several values are consistent with simple relations. We must use the "geometric HaHa trick", or deeper calculations, to get a contradiction or a direct proof.