Ok.
So, why mounted archery? Mongols are awesome, yes, but this makes no sense. If you were a scout/ranger, it would, since skirmish is gained by moving. But as a rogue, you basically need stealthiness (good luck on a galloping steed or even a winged bat/bird), flanking (a valid use for the companion, but not if you're mounted on it; other than using it as transport, then quick dismounting), or ring of blinking (n which case, you'll probably just fall off the freaking horse).
Still do your build, I would just change the focus a bit...
Mounted Archery enables me to have mobility and full attacks at the same time (and it looks like I'll be able to flurry with my bow, so with Rapid Shot that's six attacks per round, isn't it?) which should be a useful ability. Setting up sneak attack might be tricky but I figure it's nothing I can't get around with greater invisibility for me and the mount. And I'd rather have the option of sneak attacking or fighting mounted than have only one or the other on the table.
Bloodline: A major bloodline is +3 LA, yes? Unless your DM is using buy off rules different than in UA, the levels you need just to START buying off climb very quickly. I would stick to LA +1 at most if so. I'm not that familiar w/ bloodlines, but doesn't one of them (Titan?) let you use oversized weapons at no penalty? Bow damage is based on the size of the bow, not the arrows, so that can help up your damage. Obviously if you're doing a small, sneaky race disregard, this would be if you do human or some other medium creature.
Race: I'm a huge fan of Wood Elf from the MM. +2 Str and Dex, -2 Con and Int. The Con hurts, though. Wild Elf is just +2 Dex, -2 Int, might be better. If the DM allows stacking the fire race stuff onto an elf racial variant, you can negate that int penalty with a Fire Wild (Wild Fire?) Elf. Your DM's rules for racial HD confuse me, it sounds like he's assuming they all have the same amount of LA. Some only have 1 or even none, some have a bunch of LA. You definitely want a race with darkvision or at least low-light. SA is screwed by miss chance, poor lighting is a common reason to encounter miss %.
Bloodlines don't quite work like LA. I get the benefits, but have to pay some XP at certain points to keep getting them. Titan does let you use oversized weapons, or at least an oversized hammer, but I don't know how comfortable my DM will be with that (he retains the right to veto choices, of course). Beyond "Use Oversize Weapon" I'm not sure that Titan gives me any benefit that Storm Giant wouldn't. As far as Race goes, I would like to pick up Low-Light Vision, or better yet Darkvision, but there aren't many choices for that in core. I could conceivably go Dwarf, or Orc, but they've got their own drawbacks. I'm not sure that Dark Vision from either of those would offset said drawbacks.
The RHD thing is that the only races allowed are those with 1 HD and 1 LA or 1 HD and 0 LA. So if you've got an LA 1 race with 2 RHD then they wouldn't be allowed.
Flaw: I'd take Poor Reflexes. You'll have massive dex and your multiclasing gives you a base reflex of +6 by 5th level. If you do take Murky-Eyed, you need a Seeking bow even more ASAP than a normal rogue archer.
I dunno about Poor Reflexes - how often do enemies with miss chance come up? I recognize that I'm gonna have a good Ref save, but having what is essentially a -1 base save for the first few levels feels like it might be a bit of an issue. And since a Seeking Bow would basically negate Murky-Eyed at higher levels (and is beneficial anyway), whereas Poor Reflexes would still be present (and I'd have to dig myself out of the -3 hole I'm in, with whatever I do to boost my Ref save), I'm not convinced that it makes sense in the long term.
Classes: Unless you left out a detail, DEFINITELY go rogue first for the 8 skill points, I'd value those more than the 2 hp. However... what did you mean by, "I lose out on two HP and my Fort save that way"? If he houseruled that you only get the +2 for base saves from your first level...definitely go ranger. Rogue base saves SUCK. Otherwise, yes, I'd go Rogue 1, then at least one in ranger, then however you like. Going to rogue 4 instead of just 3 for Uncanny Dodge seems like a good idea. Getting to ranger 2 quickly for rapid shot also seems like a good idea.
You may want to forfeit the monk AC early on. Even with a +4 wis... chain shirt is +4 AC, and bucklers specifically allow you to use them without any penalties with a bow. You won't be proficient with bucklers, but masterwork makes that not matter (no acp to penalize you with). So you'd actually have better AC at levels 2-3 and maybe higher, forgoing the unarmored AC. Of course, no armor also gives you flurry and fast movement, so eventually you'd want to shed the armor.
He hasn't houseruled that you only get the +2 at first level, I just meant that my Fort would be lower at first level as a Rogue than a Ranger. So yeah, Rogue 1/Ranger 1-2/Rogue 2-3 seems sensible for the first five levels. As to Uncanny Dodge, if I do go for mounted combat, then I don't know that I'm going to get flanked terribly often. And it'd be the Uncanny Dodge of a fourth level Rogue, and I'm not sure that makes better sense than Ranger's Hide in Plain Sight.
As to the Monk features trade, I don't think I'd be able to pick that up after picking up my first Ranger level. Yeah, my AC may be higher for a few levels, but in the long term it would give me less benefit than the Monk trade. I could conceivably use light armor and a masterwork buckler (once I can afford it) until high levels, if that's what you meant, and I figure I'll want something for first level, but I don't know that I want to shell out for a Chain Shirt. I'll consider it, though.
Feats: *Your DM is ok with you using TWF and flurry? Even if he is, for a while that's a lot of attack penalty. I would just rely on unarmed flurry to handle melee and use a bow and focus on bow-related feats. Even with more feats, it's still a burden to get them all early on. You want PBS, Precise, and Rapid Shot (from Ranger), then obviously the Swift...Backstabber? and Wild Cohort feats. I would also be sure to get Power Attack for when you can't SA, though it's not a priority. Then maybe start thinking about the TWF line or Improved Initiative.* Even just using your mount as a taxi (or as a mount to deal with flying enemies; get a flying mount), Mounted Combat is probably worth it eventually. Mounted Archery never is. It's core only, so feats you really need will eventually dry up, more reason to not be a human.
*I write this not sure how you're getting most of these for free. You said BARB gets both combat styles, not ranger. And that still doesn't explain where some like Power Attack are coming from.
Yeah, my DM is ok with me using TWF and flurry together. And it's only a penalty if I need to - I figure a reach weapon will generally allow me to dictate whether I use it or not. I guess I could see that Mounted Archery isn't super useful, but it's still (effectively) a +2 to hit while mounted (and double moving), which is at least better than weapon focus. It's not a priority, sure, but since I've got plenty of feats I'm not sure it's entirely useless. I'll just want to pick up useful things before I get it. I may even want to pick up Ride By Attack and Spirited Charge, if I'm running out of useful feats. A lance charge for when I can't shoot can get pretty nasty, as I understand it. And I do believe that I did write that both Barb and Ranger get both combat styles (though I fear I've made the Ranger possessive rather than plural, I'll have to fix that), so that should explain TWF and all that. Endurance and Track come from Ranger, Power Attack and Cleave come from Major Storm Giant Bloodline. And Human isn't just for the free feat - the bonus to skills means I basically get +2 Int for free (at least for determining skill points).
Gear: As a rogue, you want a seeking bow ASAP, for reasons mentioned above. You also want a ring of blinking. Once you have this item, you will always be able to sneak attack with a full attack from now on, save for the occasional round to re-activate it. Since arrows leaving your possession lose any of your magical effects (size, invisibility, etc...), they cease to be blinking and thus you suffer NO chance of them blinking out and not hitting.
Ok, that's useful. If Wild Cohorts can share spells (like an Animal Companion) can they share the effects from magical items? A ring of Blinking would then mean that the horse and I both blink, and a ring of Greater Invisibility (I wouldn't need both, of course) would mean that we're both invisible, right? Or would I need one for each of us, or something like that? Are there any other enchantments that I'll want?
Skirmish doesn't let you get skirmish damage while mounted...
'Cause that makes sense. Not saying that you're wrong of course (you're not, as far as I remember), but man, WotC made some silly decisions, didn't they?