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 create a cleric, the most important question to consider is which deity to serve and what principles you want your character to embody. Appendix B includes lists of many of the gods of the universe. Check with your DM to learn which deities are in your campaign. Once you've chosen a deity, consider your cleric's relationship to that god. Did you enter this service willingly? Or did the god choose you, impelling you into service with no regard fo your wishes? How do the temple pirests of your faith regard you: as a champion or a troublemaker? What are your ultimate goals? Does your deity have a special task for you? Or are you striving to prove yourself worth of a great quest?
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:
If you’re playing a cleric or a character with the Acolyte back
ground, decide which god your deity(sic?) serves or served, and
consider the deity’s suggested domains when selecting
your character’s domain.
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.