You are confused. In this case "when, how it is done, and why" is secondary.
First you have to determine what KIND of worker we are talking about. (Article 4, Labor Code).
You are explaining the process for a REGULAR WORKER, but, as I said before, Housekeepers (I'm assuming the person is working in a private house not a hotel) are regulated under specials provisions in the Labor Code as I mentioned before, as "Trabajadores Domesticos" or "DOMESTIC WORKERS", and :
THEY DO NOT RECEIVE BENEFITS PROVIDED FOR GENERAL WORKERS.
THERE'S NO PRIOR NOTICE or SEVERANCE PAY
DOUBLE PAY IN DECEMBER IS NOT MANDATORY (it is customary, specially if you want them to return after their vacation!)
YOU DON'T NEED TO NOTIFY THE SECRETARY OF LABOR (can you imagine the chaos if every domestic worker in this country has to be reported to them?!)
YES, YOU CAN FIRE THEM BECAUSE YOU FEEL LIKE IT. (of course, ethic is an issue here).
I'm telling you this as a first as a lawyer and, second, as a Dominican with all my life dealing with domestic workers.
Ask any Dominicans, if they pay "liquidaci?n" or severance pay to a "trabajadora" or domestic worker, and you'll see.
If you know of some foreigner that paid this, well, let me tell you that they were fooled.
If you speak Spanish, check the Labor Code at the link (Articles. 4 and 258 to 265) and you'll confirm my statements. If not, let me know and I'll translate them for you.
SOME ADVICE:
Don't pay them with your company's checks, they can use it to try to prove they weren't domestic but regular workers with more rights, and they can sue you and your company. Pay them in cash, and prepare an invoice for them to sign stating the date and the concept. (they sell invoices at bookshops here, they are called "recibos") If they can't sign they can put their fingerprints.
Ask for reference. If you know people here, ask them to recommend someone. (if not, use a company that provide domestic workers).
Ask them for a photocopy of their "Cedula", (both sides) in case something is missed, or some other problem, you?ll have their data at hand.
Try to hire older than 18, the cases I've seen lately of domestic workers are when they are accused of stealing, they try to defend themselves accusing the male boss of sexual harassment, and it is worst if they are minors. (The best way to handle this is to settle, retiring the accusation of stealing, So, Keep your valuables well guarded, just in case.)
PD: Hotels' workers are not considered Domestic workers. Neither are People working for Condo management.
ONLY workers in private houses.
In general, domestic workers will respond as you treat them. And you can find here people that will work loyally with you for life.
Good luck!