Hi Detty,
As Aritz said, once you have been here over three months (this theoretically includes time spent away, but no one checks), you should apply for the new Certificate of Residency. It repalced the Residencia (a proper ID card) a couple of years back. Your local Nationa Police Station should have in place an appointments system (or not!) for you to hand in your completed form (with necessary payment slip done at a bank - just a few Euros), your photos, copy of passport (plus you take your original on the day when the officer interviews you/processes the application). You then get a green certificate (no photo), which is actually pretty useless as you will still then have to carry some kind of photo ID! But legally, if you are living here full time (and over the 3 months a year limit), you should still go and get one.
If you do not speak sufficient Spanish there are sometimes volunteers at the Police Stations, but most people pay an interpretuer to go through the process with them if they cannot keep up to speed or the volunteer happens not to be in on that day.
The British Consulate website have all the info you need about living in Spain/Legalities.
(I remember there is a dedicated Ex Pat enquiry office in Alicante. Give them a call?)
Likewise, wouldn’t hurt to go to one of their open days (they are free and usually happen once or twice a year).
Also look out for other Ex Pat organisations that offer info/support through the red tape. (Age Concern your way).
ps - The charity organisations may also be able to advise on your medical Q’s. I understood tho that if you are in paid job (no matter how low the pay), the ‘big bonus’ is that you are 100% entitled to the same medical services as a Spaniard. Private medical insurance here can be astronomical (esp if you are not earning much). Many medical centres have volunteer interpretuers if your Sp is not up to it yet.
Good luck.