You seem to be asking more than one thing here.
I can cite an example right off the bat that is in A min, starts on A min and ends on B maj chord. Luis Millan 6 pavanas and a fantasia (pavan #1). Of course the very next pavan starts on B maj so...
CORRECTION: pavan #1 starts on A min and ends on A maj (not B maj). Sorry.
You absolutely do not have to start and/or end on the chord that represents the first degree of the key you are in but that generally sounds best. As an art form there are no "rules" but in fact music is very mathematical and in Western music we have come to appreciate certain chord movements more than others. There is a definite feeling of completeness when a song starts and ends on the chord of the key (more so for ending, many songs start on the V or other chord but end on I). However, ending on another degree can be useful for expressing another feeling.
The basic chords that exist within a key make a natural cycle called the circle progression, that moves in 4ths through the key starting and ending on I.
I - maj
IV - maj
vii - dim
iii - min
vi - min
ii - min
V - maj (add 7 for resolution to I)
I - maj
Of course 7th may be added. With more knowledge of chord substitutions one can see that this is really a cover (or extension) of the basic I - IV - V progression. Once you get the circle progression in your ear you almost can't stop hearing it. It's present implicitly in a lot of music even if the players are not literally playing the exact sequence of chords. As for learning more about modes, I think it's a great idea but not sure that will shed light on your question. All seven diatonic modes are related to the major scale starting on different degrees on the scale.
Ionian (Major) starts on 1 (or Do)
Dorian starts on 2 (Re)
Phygian on 3 (Mi)
Lydian on 4 (Fa)
Mixolydian on 5 (Sol)
Aeolian (Natural Minor) on 6 (La)
Locrean on 7 (Ti)
There is another set of altered modes based on the melodic minor scale. But the point is, they are all related so once you understand the structure of western music you don't really need to have all the hole tone - half tone patterns memorized.
In my opinion one of the more important aspects of Western music is the relation between chords in the scale and the scale tones or degrees that act as a root. A traditional approach to harmony theory would point out that all notes in the major scale can be harmonized or covered with just 3 chords, the I, IV and V. Also, the concept of a resolution or cadence is a corner stone of the western musical tradition. This is the sound of V7 --> I. This is such a "strong" sound that give one the feeling of completeness that it is very common for songs to end this way. But again I must stress that this is a component of classical harmony theory and is based on cultural tastes. Not a law of physics which cannot be broken.
As far and identifying all the modes on the "white keys" of a piano. Using the formula I gave above you are in the key of C. Play D to D and that is Dorian, E to E = Phygian, etc.