As for #1 - you can either tie the 1/8th note to an 1/8th note (as you have in the stems up part of the 1st example) or you can define the space with a 1/8th rest as you have in the second example. Best rule of thumb I can think of for deciding which is to sing the music that goes with that part - in other words, how does everyone else play that "& of 4"? Do they play it short with a space between it and the note on the "& or 1" - or - do they play it as a long accented note? More than likely it will be best to write it for the drummer in a way that reflects what they are doing... even if the percussion note (tom, snare or bass drum) being played has no ability to change its duration. Cluing the drummer into what others are playing and how they are phrasing it is always a good idea.
2. Yes. This is actually the fatal flaw in both of these examples - making them at the very least, extremely difficult to read, let alone sight-read.
In any stems up/stems down part - be it drum set or a piano or violin part - promoting rhythmic comprehension ALWAYS takes precedence over any general rules as to which and where stems go up and go down.
Harking back to the early days of marching drum parts, drum set parts are most often written in a "two part" divisi style - meaning one parts stems up and one part stems down sharing the same single staff. This usually means grouping all of the notes being played into two rhythmically complete parts. This is often "hands up and feet down" - but can also be "bass drum/snare drum down and HH/Cyms up".
And depending the notes being played, oftentimes one section of the music works better grouped one way and other a different way... this is perfectly fine and pretty much standard practice. But at any given time, each group - stems up and stems down - must be rhythmically complete and readable as it's own entity.
Below are what the first two bars would look like grouped in two different ways...
One final thought - with drum set being intended be read by a single player - the rules are pretty fluid - with the focus all about being as readable as possible with as little clutter. So for instance if after a full set section, there's a section of just solo snare drum or HH 16ths for eight bars, there's no reason to slavishly write in all the whole note rests for stems down parts... as they aren't really separate parts being played be separate people, but part of one part for one player.
Yet another "final" thought - I know of no notation software that wants to write good drum parts without being forced, tricked and finagled into doing so. It is literally the worst thing they do. Why? Because the stems up/stems down aspect of drum set parts, unlike most other parts, follow a very involved and highly subjective set of "rules" - so besides the fact that they require the use of multiple layers per staff (like in divisi part), they don't follow any simple rule for doing so.) As a result it takes many, many times longer to notate a drum set part really well compared to say a bass or saxophone part.
Hope that helps,