1. Dear Elena, I regret to say that the Home Office take the view that if you have dual UK Bulgarian citizenship, then you have to apply for a spouse visa based on your british nationality. It would be possible to challenge this view by way of Court proceedings, but as the UK will have left the EU by the time these come to be heard, you might find taking legal proceedings to be a bum result. Accordingly, you are better off to simply apply as a British citizen and have your gambian husband come to the UK as the spouse of a British citizen. I know this involves a lot more hassle and the expense of an additional visa fee. However, your Gambian husband is likely to be refused an EEA spouse permit simply because you have UK citizenship as well and you are perceived as circumventing the laws applicable to British nationals.
2. This is a totally different question. There will be a question of where he is residing if he has a Bulgarian identity card. So, whilst he is likely to get past the Border post, you are more likely to run into problems internally in the uK IF HE is stopped by police or needs access to hospital care or benefits. Then the question will arise as to where he is residing. Strictly speaking, he can only live in the UK if he has a UK identity document. A Bulgarian document allows him to exercise his treaty rights in the uK ie look for work, work or student. But it doesn't give him the right to live and reside in the UK. However, whilst this will not immediately apparent at a border post. However, if he deals with the authorities in the uK, THEN THIS question will arise.
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4. Be aware additionally with the UK LEAVING the EU, this option to have a Bulgarian identity card will soon evaporate. So you will get at the most two years out of it.
5. No, if you spend five months in Bulgaria, Britain might have changed the rules as it is leaving the UK.
6. I would advise you to apply for him to be the spouse of a British citizen. As the UK is leaving the EU any other option is likely to end after less than two years.