"Dragonborn of Bahamut" are part of the Races of the Dragon splat book.
The dragonborn children of Bahamut are a unique race in that they are
not born; they are reborn. Each one enters the world as a halfling, an
elf, a human, or a member of some other humanoid race with all that
race's propensities and traits. Bahamut beckons to his would-be
followers, and those few who might choose to serve him.
The dragonborn does not become a "kind" of dragon, but is gifted with dragon-like attributes by the Good-aligned dragon god, Bahamut. You may choose which gift to claim but they are not type-specific. They do take on a draconic appearance, though to a primarily cosmetic degree. The template is a racial replacement, not a class replacement, and so does contain special bonuses but their comparison to a "spell-slinging rage wizard" is largely irrelevant as the two are not mutually exclusive.
The following article is available via Wizards' website but you should consider purchasing the book for more complete information: http://archive.wizards.com/default.asp?x=dnd/iw/20060105b
I have never before heard anyone suggest that contemplative would be so restrictive for clerics of an ideal; in every discussion of the class I have read (and it comes up a lot, being a rather convenient source of a bonus domain), the interpretation is that the bonus domain must be one you could have otherwise chosen as one of your original cleric domains, i.e. one available from your deity or faith, alignment domains can only be chosen by clerics of the matching alignment.
That said, you highlight a discrepancy with that interpretation; it does say a “domain made available by her [...] alignment,” rather than a domain made available by her faith. Strictly speaking, the only domains that your alignment makes available are the alignment domains. So by strict RAW, I believe you are correct, you can only choose the Good or Law domain since “your deity” (that you don’t have) doesn’t make any domains available, and your alignment only makes Good and Law available.
However, personally, I find this an unecessary limitation; yes, contemplative is rather popular for the bonus domain, and yes, that’s an extra feature for an already-powerful class, and yes, it is very easy to get, but meh. The cleric is one of the most powerful classes in the game with or without this ability, but the option of dipping contemplative for an additional domain makes a wider variety of characters possible that otherwise would not be. I call that a good thing, so I would allow it.
It looks to me that the authors of contemplative either forgot that clerics of ideals existed, or assumed they’d implicitly substitute their ideal for “deity” in the rules text. I base this assumption on the very fact that Law is the only choice available to an LN cleric of an ideal under the strict-RAW interpretation, which would make the 6th-level bonus domain completely pointless; when an option might be completely pointless, I expect the text to at least recognize that possibility with “if any” or some such.
Best Answer
1. Is there any requirement that the alignment be the same or close to their deity's?
No, your alignment does not need to match the alignment of your deity.
In the introductory paragraph for the cleric it reads (page 35 of the Player's Handbook):
As you can see from this paragraph, it's entirely possible for your own personal goals and aspirations to be counter to those of the deity you choose.
An example of this might be in the Tyranny of Dragons campaign: an evil deity who hates dragons might have forced you into service to fight against evil dragons.
2. Can a cleric choose a domain that is unrelated to their deity?
No, a cleric must choose a domain that is related to their deity. (Except for the Life Domain, a cleric of any deity may take the Life Domain) However! However, the list of domains in the god appendix, is not canon, but is rather recommendations. Appendix B page 293 of the Players Handbook:
3. Can a cleric choose a deity of a different race to their self?
It depends on the deity. I see no information that says you have to pick a deity of the same race as yourself. If you are asking about the Nonhuman Deities, they are examples of gods that are shared by both Greyhawk and Forgotten realms settings, and so are not listed in either. The introduction to that section also makes clear that each of the races might even have their own pantheon in which they worship. If a human grows up in Dwarven society, there is no reason for them not to worship a less well known Dwarven deity such as Haela Brightaxe. It also states that not everyone worships the same deity in the same way.
Take for example two of the sea gods listed: Deep Sashelas, elf god of the sea, and Eadro, merfolk deity of the sea. Eadro is a the creator of Merfolk and Locathah and only the Merfolk and Locatha have a relationship with Eadro, and Eadro only cares about the Merfolk and Locathah, it's more important to the worshipers of Eadro that he created him rather than his power over the ocean in general. Deep Sashelas, on the other hand is worshiped by sea elves, and land elves. He is worshiped for his influence over the sea in general, sharks, dolphins, beauty and magic as related to the sea, etc. So while only Merfolk will worship Eadro, anyone with contact and access to elven society might worship Deep Sashelas.