Say, when one's practicing the first four keys in the C-major scale, how useful is to practice it with fingers 1-2-3-1 as opposed to 1-2-3-4? I could say to myself 'what ever you feel easy', but is there a benefit in crossing over finger 1 over 3 to hit key 4? What's the suggested way? Is finger usage 'forecast' done based on how many keys the pianist plan to reach at a given moment?
Piano fingerings are all about context. The notes which come before and after dictate the best fingering now.
in the case that you're playing the 1st 4 notes of C major it depends what the 5th note is. if the note is the root C again, then the thumb is probably the worst finger you can use for playing F, if on the other hand you're continuing up the scale over multiple octaves, the 12312341231234 is more likely to be the smoothest way to go. The value in practising crossing fingers only really comes in that you train your fingers to find the most efficient ant practical way to play any piece.
I'd say that in most real pieces there's not a direct scale from root to root, but practicing scales using finger crossover allows your fingers to develop the technique to cope with runs which you do find in pieces.
A great book I'd reccommend for developing a good grounding on finger technique is Schmitt's 'Preparatory exercises for the piano' That to me gives a great sense of piano fingering problems at their simplest.
I'm learning while playing bach that context is the key to piano fingerings. what came before these 4 notes, and what is coming after dictate what fingering you use now.
Is there a benefit in crossing over finger 1 over 3 to hit key 4?
Learning evenness in crossing fingers allows you to move more freely around the piano, and allows you to make the transition from 123 to 12
What's the suggested way?
It entirely depends on the context, as in the notes that came before and after(and possibly the style you're playing in)
Is finger usage 'forecast' done based on how many keys the pianist plan to reach at a given moment?
It's based entirely on the most practical and efficient way to play the notes, in the intended manner of the piece.
Yes, there are times when it is beneficial to 1-2-3-1 instead of 1-2-3-4. Let's think of the concept this way if your melody is in C major and uses only 5 notes from C to G then you can have one finger on each note and be fine. However, if your melody is in C major and uses 8 notes from C to C (an octave) then you need to be able to move your fingering to reach the other notes so if the next few notes are higher than your hand is currently positioned you need to shift over somehow and the crossover from 3 to 1 may help. The range of a piece determines how you should finger each note. If the piece has a large range (an octave or higher) then you should look for ways to shift your fingers while playing.
Think of it this way, one of the first things any pianist learns are scales and in every scale you do some type of finger crossover. If the melody did a run up the scale, it would be optimal to do the finger crossover which in C major and most sharp keys is 1-2-3-1.
However at the end of the day it has to be comfortable to you. If any fingering is uncomfortable, then you should examine how you're playing it and see if it can be played an easier way.
Fingering should be considered very carefully. 1231234 is properly used for scales such as C major, and is indispensable for your technique in general. Using the same attitude and common sense, 1234 should be used for F major and B major in the left hand.
The point is you should not prefer one over the other, but be comfortable with both, and have no problem using them when needed in a 'real piece'.