This is quite a complex question. Because it does not only takes into account the character stats block, it also take into account general rules and how they interact with specific rules.
On any RPG, or any game, actually, you should always apply the general rule, except when a specific rule say otherwise.
So what is the general rule for bleeding? You need to be alive, and need to depend on blood to be alive.
What the general rule say about suffocation? The text is as following: "When you run out of air, you fall unconscious and start suffocating. You can’t recover from being unconscious and must attempt a DC 20 Fortitude save at the end of each of your turns. On a failure, you take 1d10 damage, and on a critical failure, you die. "
So, you become unconscious, and you have to roll fortitude save or take damage. If you critical fail, you die (i.e., take a DEATH effect).
What the specific rules of undead say about those things?
"Once living, these creatures were infused after death with negative energy and soul-corrupting evil magic. When reduced to 0 Hit Points, an undead creature is destroyed. Undead creatures are damaged by positive energy, are healed by negative energy, and don't benefit from healing effects."
Therefore, you are not a living creature. That means that by RAW, any effect that target living creatures can't target you. Bleeding, by raw, only affects living creatures.
But wait, the basic undead rules say that player undeads are different, right? So on what way are they different? Only on those aspects that are written on the basic undead benefits block. What does it say about living?
" Because you're undead, many methods of bringing someone back from dying, such as stabilize, don't benefit you. When you would die, you're destroyed rather than dead, just like other undead.".
So, you are not really alive. You don't die. You are destroyed. What does that mean? You can't be resurrected, for instance (since the text contains the string: "The target must have died within the past year").
So, no, you are not a 'living dead'. You are an undead. And you, by RAW, are not afffected by bleeding (nor ressurrection, reanimation, or any other way that could bring you back to life traditionally).
Alright. But what about breathing?
The question is not if you need to breath. But rather, are you affected by suffocation?
Suffocation makes you unconscious, force you to take bleeding on a failed save, and and you die if you critical fail.
So, can you become unconscious? There is nothing on the undead or basic undead rules stating you can't. By RAW, if the undead (even a monster) don't have a specific rule saying it is immune to unconscious, it is not.
Can it take damage by suffocation? Again, that is not any rule saying that an undead does not take it. So, by RAW, all undead should take damage from suffocation, except if is said on it's stats block it does not.
Can it die from a critical failure? No it can't. Both the undead trait and the basic undead benefits block say they are immune to death effect.
But... the suffocation only triggers after you run out of breath.
The only rule about breath I found states:
"You can hold your breath for a number of rounds equal to 5 + your Constitution modifier."
So, there is no actual written rule stating that undead breath or does not breath. Just that, if does breath, it can hold their breath for a number of rounds etc.
So, what is the general breath rule? None. What is the specific breath rule? We only have one for aquatic and amphibious creatures, stating that they can breath under water (and that aquatic can't breath air). Also, some planar creatures (such monitors) have on it's traits they " can survive the basic environmental effects of planes in the Outer Sphere". Meaning they can breath there, and don't suffocate if they normally would.
There is no rule on undead that state they don't have to breath. On incorporeal, on the other hand, it is said: "Incorporeal creatures usually have immunity to effects or conditions that require a physical body". By definition, only those with physical body breath (incorporeal does not interact with air, since air is corporeal). So ghost are immune to suffocating not because they are undead, but because they are incorporeal.
So, to make long story short:
(a) does skeleton take bleed damage? By RAW, they don't, since they are not living (requirement for bleeding effect).
(b) does skeleton breath? By RAW they do, since they are corporeal creatures with no special breathing stats.
