Closing with "Sincerely" and your name might work, or with your name and title without a closing phrase.
Yes, you should capitalize "To Whom It May Concern" in a cover letter.
To whom this may concern
to whom it may concern
When you do not know who will be receiving the letter.
'To whom it may concern' is right. 'To whom soever it may concern' is wrong.
Yes. It should be --- To Whom It May Concern
To whom it may concern
Whether in writing or in person, address a judge as "your honor". Do not use "to whom it may concern" unless you have no idea who will be reading your letter.
Address it "To whom it may concern"
Yes.
To Whom It May Concern.
"Can you write a letter of recommendation for me?"