You ask if Arminians are under the law. In the sense that all people everywhere owe all that they are to God, the maker and creator of all things, then all are obligated to live completely and fully for Him. That then is the requirement of God. But only Jesus fulfilled the requirement. Therefore all the rest of us are condemned for our sin against God. But through Jesus, we who trust in Him have been set free from the Law. That freedom is not a license to sin but a freedom to live and love God apart from condemnation. Arminians who fail to accept that freedom [and that would be not all of them, some Arminians believe in eternal security] put themselves back under the law.
Now if they are saved, it doesn't mean they lose their salvation, but what it means is that they are arguing against salvation by grace and for salvation by works. If they persist in that anti-gospel position, i would begin to doubt if they were ever saved to begin with.
