@Lauren_123
Welcome to the expat.com forum, and good luck with your potential move to Spain!
Yes, you're quite right. EU citizens enjoy "freedom of movement" and can move to any EU country, for any reason. He just needs to complete the immigration formalities for "EU citizen registration" which typically requires proof of address (e.g. LT rental contract), proof of funds, and proof of health insurance.
Official Spanish immigration page on the process is here:
https://extranjeros.inclusion.gob.es/es/InformacionInteres/InformacionProcedimientos/CiudadanosComunitarios/hoja101/index.html
Once this is approved, and they issue the certificate, he is a legal Spanish resident. He can then immediately apply for "family reunification" for you as his family member. The main requirement for family reunification is proof of your family relationship (e.g. civil partnership), and your own proof of health insurance.
Again, the official immigration page is here:
https://extranjeros.inclusion.gob.es/es/InformacionInteres/InformacionProcedimientos/CiudadanosComunitarios/hoja102/index.html
Therefore, in answer to your questions...
1. No visa required, your UK passport gives you up to 90 days in 180 visa-free. Plenty of time to follow the 2 steps above (or, if you're worried, send him ahead to get his registration certificate, and you come over when it's approved for step 2 only).
2. You can start researching where you want to live, perhaps even organize your rental apartment, and study the above information. But there's very little you can do in the UK to start the above processes, which are done in person at Spanish immigration offices. Each process requires an appointment and this is done online via the cita previa system, so you could make the appointment while in UK. There might be translations and certifications ("legalisation" in London) required (maybe your civil partnership document), so this too can be done before arriving. Maybe health insurance with Spanish healthcare company can be organized. But you can just rock up in Spain and do it all there. :-)
3. I'm assuming, yes, but I don't know. I know the EU-wide immigration rules are actually for couples who are legally married OR in a registered (civil) partnership OR unmarried but cohabiting in a "durable" (more than 2 years) relationship. We were in the last category, and Spain was a bit difficult about it, but they did approve us.
4. Making an appointment and going to immigration to submit the relevant application is pretty cheap. If you take a Spanish translator to help, that won't be a massive cost. If you ask a lawyer to do it, that might be a grand or so. Otherwise, the costs are in the proofs. You need an address, so you gotta buy somewhere, or rent somewhere long-term (better in joint names so the proof works for both of you). Proof of funds needs a few thousand (I can't remember the exact number, but based on Spanish minimum wage, and the above pages have the up-to-date info). But, say, 5k-10k in a Spanish bank account (ideally), or perhaps a UK bank account. (But if he gets a job in Spain then his contract/salary is proof of funds instead.) You can spend it after approval, so it's not strictly a "cost" of the process. Spanish health insurance can be 500-1,000 euros say, per person, so that's a good chunk.