I assume Yes, but can't find an answer in Pathfinder rules.
There's two different questions here - what conditions in general "don't stack" and you just take the worst, and which conditions "double down" when reapplied, and then which one of these ways does fear work? Fear doubles down; in other words it stacks and worsens down the condition track.
They don't list it in the condition summary, but under the Fear section you can see that someone who is shaken that gets shaken again moves up to frightened, etc. That's a special sort of stacking that only fear and a few other things do. Note that in several places, like the Intimidate skill, they deliberately nerf this by saying "two shakens caused by Intimidate doesn't stack to frightened" - they only say this because usually they do.
So yes, specifically, if you Intimidate someone and they're shaken, and then cast Doom on them and shake them again, they are frightened (not panicked, you'd need another for that - a second Doom could do it, but not another Intimidate because Intimidate explicitly doesn't stack).
Normally conditions don't stack, but this does require interpretation at times. You don't Dazzle someone twice and then then become Blinded, but also they don't explicitly say Dazzled is a lesser condition than Blinded. In general it's easy when it's all numerical - "-2 morale penalty to attack rolls" and "-1 morale penalty to attack rolls" nets you just the -2. It's "obvious" that dazzled shouldn't stack with blinded even though the penalties aren't technically of the same type (-1 penalty to attack vs 50% miss chance) but that's something you/your GM just has to be smart about.
How does the lighting effect the dice roll?
How does vision type impact the roll when its not full day light?
That depends on how dark it is, and what kind of vision they have. Here's the rules for vision and lighting:
In an area of dim light, a character can see somewhat. Creatures within this area have concealment (20% miss chance in combat) from those without darkvision or the ability to see in darkness. A creature within an area of dim light can make a Stealth check to conceal itself. Areas of dim light include outside at night with a moon in the sky, bright starlight, and the area between 20 and 40 feet from a torch.
In areas of darkness, creatures without darkvision are effectively blinded. In addition to the obvious effects, a blinded creature has a 50% miss chance in combat (all opponents have total concealment), loses any Dexterity bonus to AC, takes a –2 penalty to AC, and takes a –4 penalty on Perception checks that rely on sight and most Strength- and Dexterity-based skill checks. Areas of darkness include an unlit dungeon chamber, most caverns, and outside on a cloudy, moonless night.
There would be a -4 penalty if you're in darkness, unless they have darkvision. There isn't a penalty in dim light. True darkness is less common than you'd expect at the ranges you're talking about, as someone in the party likely has a light source (unless they all have darkvision).
Low-light vision increases their light range so that they can use a light source that's farther away for dim light. There's a chart in the linked rule that has more details.
Note that if you're attacking someone in dim light or darkness and you don't have darkvision, you can't Sneak Attack because the target has concealment as well. From the Sneak Attack class feature:
The rogue must be able to see the target well enough to pick out a vital spot and must be able to reach such a spot. A rogue cannot sneak attack while striking a creature with concealment.
Do his friends get to roll to see me and warn him somehow during my approach?
They could get to roll Perception to see you if they're in a position to do so. They do not get to warn him or take any action as this is all happening on your turn, unless they somehow have a way to act on your turn (such as a Ready Action). Technically speaking they can speak on your turn, but I can't remember any DM who ever allowed that as a defense against stealth attacks. The target can't react as it's not their turn, and they still can't tell where you're coming from as they failed to notice you. Knowing there's someone somewhere in the area who might attack doesn't negate the surprise of an attack when it actually comes. (If I did, "I always expect an attack" would basically negate invisibility's surprise.)
Note that after you attack you lose Stealth anyway, so a roll from the other characters except the one you're attacking is likely a waste of time.
If its full day light can I still do this?
This appears to be highly contentions in the Pathfinder world, but as far as I can tell, no. Here's the Stealth rule:
Breaking Stealth: When you start your turn using Stealth, you can leave cover or concealment and remain unobserved as long as you succeed at a Stealth check and end your turn in cover or concealment. Your Stealth immediately ends after you make and attack roll, whether or not the attack is successful (except when sniping as noted below).
Now, here's the problem (again from the vision and lighting rules):
In an area of bright light, all characters can see clearly. Some creatures, such as those with light sensitivity and light blindness, take penalties while in areas of bright light. A creature can't use Stealth in an area of bright light unless it is invisible or has cover. Areas of bright light include outside in direct sunshine and inside the area of a daylight spell.
Bright light means you're not in cover or concealment (without getting it from another source). While the rule says if you started in stealth you can move and you lose it when you attack, the rules also say that you can't use the Stealth skill at all in bright light without cover. If you step into the open to attack you don't have cover, which means you can't use Stealth at all. Since the target is allowed a perception check when it's most favorable to them (see the designer note), they get to make it once you're in the open and automatically succeed.
In that case you'd need some other means to grant the conditions to use Stealth. Some options are spells such as Invisibility, Deeper Darkness, or Fog Cloud, along with feats like Hellcat Stealth.
That's the opposite of what I said before, but given the way that Perception works I think it's more correct. Once you step into bright light your Stealth doesn't work as per the light rules. That means they see you. The attacking from Stealth rules work if you're attacking at range from behind cover, or you have conditions that let you use Stealth.