Hola Zyrafa,
Your situation is not unique and your employer will surely be willing to keep on doing what htey are by not actually commiting to your full time, full benefit employment.
Seems like a way of stringing you along to me and it happens often here, especially in the English tecahing field.
They will say "don't worry" but it is in your interest to get them to make you legal and give you papers for a long term stay here.
They probably have you on their books (if you are on their books at all) as a "becario" ("intern") which is the way many companies can pay you while you are not really in a legal working situation here.
That is ok for a while but as I said, get them to do your paperwork and get you residency if you can.
I have a friend who was working here in Madrid as an "intern" at a commercial production comapny and the boss was telling him over and over not to worry about papers. He was getting paid a decent wage for an intern but wanted papers. After 8 months of "interning" he wanted an answer on the papers thing and insisted. He was then told that it was not going to be possible.
En fin, your employer is getting a lot out of having you work the way that you are and you are as well since you are able to work here and live in AMAZING Madrid while you study at the same time. Also, you got a gig in your field and not in English teaching which is what 95% of all native English speaking ex-pats end up doing here so you are lucky but GET THEM TO WORK ON YOUR PAPERS WITH RESULTS. They most likely have lawyers on staff who can do it all for you.
As for finding a friend as a fiscal double, it would have to be a really good friend since that would imply quite a bit of paperwork for him/her as an "aut�nomo" and he/she would alos have to trust you to pay him/her for the taxes that he/she would be paying for you.
You are much better off trying to get the company to make you a legal employee.
Saludos,
jer...
p.d. Don't worry, I do not think you will have trouble coming back to Spain after your holiday at all