@cogaho9887
Welcome to the expat.com forum and good luck with the move to Bulgaria!
It's a great country, I'm very happy here. It's especially good for digital nomads / remote workers because of the low cost of living and low taxation (10%).
And, lucky you, it's easy-peasy for EU citizens! You can work, study, retire (or hang out on the beach) in any EU country thanks to EU "Freedom of Movement". It's recommended that you do an "EU Citizen Registration" at your local immigration office. It just needs proof of funds (I used a copy of my debit card), proof of health insurance (either your French EHIC or a cheap Bulgarian Health Insurance for Foreigners for about 100 euros), and proof of address (your long-term rental contract + landlord declaration). They will issue a plastic biometric residence permit and ID card valid for 5 years.
EU citizens don't need a criminal record check. And the whole process is pretty straightforward (should have your card within a couple of days). However, I think it's worth taking a helper/translator with you (mine charges about 50 leva per hour, and immigration offices have big lines unless you're up very early).
Registering in the Bulgarian social security / healthcare system is not mandatory, but I'd recommend it. I registered as "unemployed" which is the cheapest option at around 27 leva per month (employed or self-employed are also options, contributions based on salary). You can then get free (or nearly free) medical treatment, and a Bulgarian EHIC (if you want one). Most private hospitals (which are obviously a bit fancier and less busy) are integrated into the system so they get the standard state payment, and you just pay a bit extra for your VIP treatment.
For your work, you can choose to be self-employed in Bulgaria. Or, you can incorporate a Bulgarian company (around 350 euros) and invoice via this company. Most likely you'd add yourself as an employee and pay your social security this way. Your business profits are subject to the 10% corporate income tax.
I suggest you also check your French pension situation too. You might have many years of contributions already, but not enough for a full pension. Not all countries allow this, but if you can make voluntary contributions it's usually well worth considering. I'm British and I was short by 16 years or so, and I was able to top-up at a very good rate (Class 2 NICs for any Brits reading).