Common practice nowadays is to show all notes that are played simultaneously on a staff using a single stem if they start and end at the same time, and if nothing else will happen on that staff during that time. This is done even in choral music where a staff is shared e.g. between soprano and alto, or between tenor and bass. Notation is only divided into up-stem and down-stem portions at beats where different notes start and end at different times, or when there are distinguishable voices (e.g. sopranos and altos) and some but not all should refrain from singing on that beat.
In beat 2 of measure 1, bass clef, not everything is supposed to sustain a half note, but something else is supposed to happen during that time (i.e. some quarter notes are supposed to play on beat 3). Thus, at beat 2 the bass clef splits into two logical voices, one of which plays half notes, and the other of which rests for a quarter note and then plays quarter notes.
Note that on beat 3 of measure 2, the treble clef actually has three voices, with the middle one playing a quarter note while the outer ones play eighths. The bottom voice is beamed with the eighth note on the second half of beat 3 in the bass clef because doing so avoids having to write an eighth rest for the first half of that beat, and because it makes clear that that note starts at the same time as the quarter note but lasts a different duration.
Had the bass clef not had an eighth note that could be beamed to that note in the treble clef, other ways of notating that measure would be:
Notate the treble clef with homophonic eighth notes including a "d" which is tied to another "d", avoiding the need for quarter notes. This would avoid crossed stems, but such ties can make music hard to read.
Place the "a" on the same stem as the "e" and "g" above. This would avoid the need for ties, but at the expense of crossed stems.
Stack the notes as they are, but use a "flagged" eighth note on beat 3, and add an eighth rest for the second half of that beat.
Compared with the above, the use of the cross-staff eighth-note beam was a reasonable "least of evils" judgment.