First of all, there is no absolute standard for drum notation, or percussion notation in general.
The rule of thumb (usually) is:
- the position in the staff is based on (general importance order):
- height of the limb that is used to play it (arms on top, legs on bottom);
- physical position of the instrument on a "standard" set;
- pitch of the instrument;
- less common/played instruments are more distant from the staff (a splash cymbal is usually higher, not only due to its high pitch) and/or have different note heads (cross, diamond, etc.);
- cymbals and "keeping-time" instruments are on top, generally starting from the upper space of the staff; with "keeping-time" instruments I mean those that usually have a sound with a distinct attack (transient) and a very short decay, and normally play the beats or any of the subdivisions;
- snare drum is in the middle: some place it in the third space (C in treble clef), some on the mid line (B), others in the second space (A);
- bass drum is at the bottom-most, usually the lower space (F) or below the staff (D);
- the same for any instrument played with feet, as explained before (including double pedal/bass drum which usually is placed a note lower than the "main" pedal whenever required, or hihat pedal which is normally at the same "pitch" as the bass drum, but using the cross note head);
- any other headed drum is above or below the snare, depending on their pitch and their amount;
- other instruments are placed at a position similar to their pitch, usually with different note heads if the set requires different instruments (cowbell, triangle, etc);
- special usage of an otherwise "common" instrument uses different note heads or markings above or below the stem orientation: a triangle or diamond note head on the ride often means playing on its bell, a small circle on the hi-hat indicates that it's an open hi-hat, with the pedal (possibly partially) released, a plus sign indicates a closed hi-hat whenever it was opened before, or a choked cymbal (stopped with the palm), a cross note head on a snare position is normally for "cross stick" or playing on its rim;
Any of the above depends on many factors: complexity of the drum part (or setup), roles of individual instruments, experience of the copyist/composer, known understanding/experience level of the player.
It's common practice to put a legend at the beginning of the part/book, explaining what each note/head refers to.
Keep in mind that for relatively simple parts like the one you linked, even a small experienced drummer should be able to understand your writing if you follow the above general rules: no matter if you put the bass drum in the first space of the staff or below it, it's clear that it wouldn't be the snare or a cymbal.
Also consider that for very complex parts it's possible to have a staff with more than 5 lines or even a two-staff system.
Note that this is valid for most "modern" drum parts. Jazz (mostly big band) parts are a bit different, as the role of the drummer there is not as "groovy" as that concept has become in more recent times: it is/was more about carrying time and motion than "keeping the beat" as it became from later times. In fact, most big band drum parts have very few notes (sometimes they just have the "cymbals" in even-eights notation, or even just "bars" for the quarters) and the only explicit parts are for "stops", "specials" or solos. Most jazz drummers just see the structure, they don't actually "follow" the page, as their role is a bit different than rock/pop drumming: they still keep time for the whole band, but with a different feeling.
Then, the problem of stems is a bit awkward.
For simple rhythms, stems are usually pointed upwards, mostly because you almost always have "upper notes" (hihat or ride) along with "lower notes" (the groove), and it's easier to get the overall rhythm when stem groups are above (the cuts); also, the general rule is that stems are always upwards when the lower note is an octave below the higher and is at or under the first line of the staff.
BUT, depending on the situations, it's not uncommon to have stems pointed upwards for parts that only keep time (see above) and downwards for parts that play the groove, which is useful to allow the drummer understand the difference between those two "roles".
I actually prefer to have divided parts for my students, as understanding the difference (and interaction) between those roles is very important.
Using different voices for every part is generally discouraged, as it usually results in an overcrowded score that is just hard to read and doesn't provide any real benefit.
About the last part: yes, you should rely on that. Drummers are used to repeating patterns, as they don't think about their part as in a "complete score" view, but as in a schematic structure of the song, which obviously includes repetitions and variations of basic patterns. They normally count bars/repetitions and/or follow the song structure in order to know what they are ought to play.
In fact, having a score that has all bars explicitly written is actually discouraged, as it makes reading very confusing. Imagine having a section that has the drums repeating the same rhythm for 32 bars: with a standard score spacing, you'll end up having at least 4 or 5 lines that all have the same repeated pattern; as soon as the drummer looks away even for an instant, she/he will most probably be lost.
If you have lots of repeated bars, a common solution is to put small numbers above each repetition or group of repetitions, following the song structure. Placing double bar lines at section boundaries is also helpful.