1. 21% is VAT. You hourly rate must be always VAT excluded.
2. I suppose you are referring to the "cuota de autonomos". This is 330 euro/month. New freelancers can get a cut during the first 2 years (100 euro/month if I remember well)
3. Nope. If you are a freelancer, the "cuota" includes social security (this includes public health service, so you don´t need a private health insurance; and yes, public health system in Spain works really well most of the time)
4. Then, you have the actual tax for your work (IRPF). In every invoice you need to withhold 9% of the gross amount during the first 2 years, 21% afterwards. Every 3 months you need to file "modelo 130" (more or less equivalent to an additional 9% monthly withhold) and once a year the government will net all these amounts. Depending on your personal situation (married,kids,mortage,etc), you may get some money back.
Nevertheless, 5-7K/month seems pretty accurate for Barcelona.
Hope this helps