There are no rules for resolving multiple DCs some with and some without disadvantage - use multiple checks or passive checks
Example 3 is the tricky one here, but I will cover the first two anyway. You said not to use passive perception, however both of those two examples should be resolved using passive checks as per the rules.
Refresher on passive checks:
A passive check is a special kind of ability check that doesn't
involve any die rolls. Such a check can represent the average result
for a task done repeatedly, such as searching for secret doors over
and over again, or can be used when the DM wants to secretly determine
whether the characters succeed at something without rolling dice, such
as noticing a hidden monster.
Example 1
A character standing guard in torchlight while the others are sleeping:
Should they roll their perception check with disadvantage because an
enemy will be approaching through the dim light, even though other
things might happen in the bright light which should be caught by the
same check, for example a rat stealing some rations from their
backpack?
When standing guard you are doing something continuously, and should use a passive check. So in this case you don't have to make two rolls, instead you use passive perception and compare that to the stealth check (or DC if the enemy is not stealthy) of the enemy they may notice. When the enemy moves into the light the passive perception increases as the penalty is removed.
Example 2
A character wanting to look out for traps as they move through a dungeon,
where a hidden wire or pressure plate would first come into view in
the dim portion of torchlight before entering the bright area. Do they
roll normally and simply not notice anything in the dim light? Should
the DM subtract 5 from their roll for the dim area?
Normally you would use a passive check for this rather than using an active check. As the trap becomes more well lit the passive check is adjusted and the trap may be revealed.
Example 3
A character wanting to examine a wide field with moderate foliage in a
few patches for any sign of enemy activity, where a goblin is hiding
in one of the bushes and faint tracks of goblins in the grass? Should
the singular action of examining the field be divided up into multiple
rolls for each part?
This is the real problematic situation. The goblin is lightly obscured, while the goblin tracks are not. As such, you want to make a roll which covers both searching for the tracks with a normal roll and searching for the goblin with a roll at disadvantage. Unfortunately there are no rules which allow that kind of check to be made.
The problem here is the granularity of the rolls. The player is asking to do two things with one roll: search the grass, and search the bushes. By RAW you make one roll for one thing. So, break it down to two rolls; you can search the grass, then search the bushes at disadvantage.
While making multiple checks works ok for this example, it can be awkward as the scale increases. For example:
Two items are lightly obscured, one in a tree far away, one in a
barrel close by, while a third is not obscured lying in the grass.
It feels strange to ask the player to roll once to search the grass, then a second time to search the barrel and the tree since those are unrelated objects far apart. It also feels a bit excessive to ask them to make 3 rolls - or more if there are more barrels/trees in the area and you don't want to tip them off as to where the items are.
However, 5e has mechanics for resolving "a lot of rolls"; passive checks. If the character needs to search a whole bunch of places, then just use a passive check. There are no guidelines as to when you should switch from active checks to passive checks other than that they are "average result for a task done repeatedly". I would suggest that if your resolution feels unwieldly then you have passed that threshold.
Remember Stealth provides equivalent bonuses
Keep in mind the rules for advantage:
The DM can also decide that circumstances influence a roll in one
direction or the other and grant advantage or impose disadvantage as a
result.
Generally speaking being in darkness or in dense foliage is a circumstance which is advantageous to being stealthy. In example 3 above, the goblin is hiding in bushes and thus is at least lightly obscured. As such, they will be making their stealth check with advantage.
While it is true that the goblin will also be harder to perceive than the tracks due to the bushes, you should keep in mind that the goblin has already gotten half its bonus. A DM who is playing fast and loose with the rules could simply ignore the problem in Example 3 and ask the player to make a normal roll. Since the goblin hid with advantage they will already be somewhat more difficult to find than the tracks.
Personally I do not do this, as I want the goblin to be just as hard to find whether or not there are tracks. But if this situation comes up once in your entire campaign, it may be beneficial simply to make a quick and decisive ruling rather than digging deep.
