I agree with the quote from the link in #3, which essentially implies that none of the gendered options are a good choice.
Like eric, if I absolutely can't dredge up a name, I would address it to the relevant department or to a description of the person or people who will need to read it. On the envelope and in the heading you can put 'Attn: Customer service,' which stands for 'Attention.' For the salutation I would still use 'Dear' so it will still look like a letter and not a memo:
Dear customer service department:
Dear human resources manager:
Dear technical support team:
But yes, that can look a little silly, so it's often better to call or go online and find a real person's name. Even if it's not exactly the right person, at least that person will then be individually responsible for forwarding it to the right person.
If you feel absolutely compelled to use gendered language, you can certainly use 'Dear ladies and gentlemen' or 'Dear sir or madam' if you wish; they're not wrong, they just aren't used by most native speakers, at least in the US. 'Ladies and gentlemen' does sound more like public speaking than writing, and 'Dear sir or madam' can sound a bit quaint and rather cold.
So I'm skeptical about the quality of that 2004 book quoted in #10c; surely it's at best an outlier, since hopefully most people have stopped writing 'Dear sirs' in this day and age in the UK as well.
None of these terms, by the way, are capitalized, because they're not proper nouns, unlike the name of a person or a company.
I would also have absolutely nothing against
Dear CM2DD Combine Harvesters Ltd.:
but it might work better if you know that CM2DD Combine Harvesters Ltd. is a relatively small company without a separate customer service department.
And as for the law firm in New Jersey, I think you would be entirely justified in writing back to them:
PS: In reference to your salutation "Gentlemen," please note for your records that both women and men work in our company. Thank you.
Though if they're stubborn traditionalists, they probably know that and just don't care. (-: