If you are responding to an advert which named someone, as in "reply to Joan Smith/John Brown", then use their name - Dear Mrs/Mr/Ms/Miss [surname]. If you don't know the name, but can in some way divine their sex from the ad (I know that's unlikely), use Dear Sir or Dear Madam. If the ad just names the business, use "Dear Sirs". I know that is a bit sexist, but the pedantically correct "Dear Sirs or Mesdames" doesn't cut the mustard, somehow. It is common practice to use Dear Sirs when writing to an all-female outfit, and it doesn't seem to raise any hackles. But if you can find out the name of the owner by asking local pet-owners, use it - "Dear Mr/Mrs/Ms XXXX"
If you are writing on your own initiative, rather than responding to an advert, use "Dear Sirs", unless you can find out the owner's name, inwhich case use Dear Mr/Mrs/Ms XXX.
If you can use a name, it helps. People like seeing their names on letters (and a letter would be better than an email).
I'd avoid "Dear Staff Manager" - a small business probably won't have someone with that title. However, you could address the letter to "The Staff Manager, HappyKats Hotel, etc etc and follow it up with "Dear Sir or Madam" - that would be OK.
As for signing off, the convention is that if you begin "Dear Sir," "Dear Sirs," or "Dear Madam" then finish with "Yours faithfully". If you began with their name as in "Dear Mrs Smith", then use "Yours sincerely".
Print your name under your signature, and if female add (Mrs), (Miss) or (Ms) after it. This makes it easier for them to reply to you.
I think you posted another question asking about reasons for applying for the job. You and the kennels/cattery will both know that it probably isn't going to be your life's work, so why not say that you like animals and hope eventually to become a veterinary nurse, so need some hands-on experience?