I can't pretend to have full and complete answers for you. To some extent, the answers vary on a student-by-student basis. Your second question is easier to answer than the first, by far: simply have the student talk you through their solutions. It's not a bad idea first to watch the student work through the process with no input at all from you. Be comfortable with the student struggling to explain, but also be firm in requiring correct language. If a student can clearly explain a process, using correct language, then that student understands the process. To be more certain, then throw a few different kinds of problems at the student, all requiring the same concept.
As for your first question, this is, as likely as not, an attitude problem as much as anything. I'd be willing to bet that your students who have this problem think very poorly of mathematics. They don't see the point, they think they're horrible at it, they've probably heard other people say they're horrible at it, etc.
Now one thing I've learned is not a good thing to do: try to convince the student that "they'll use this in their future life." Actually, they probably won't. Unless you're teaching basic arithmetic, the chances are the student will not use it in their future life. The students know this, which is why you trying to convince them of this comes across as hollow.
So why should they learn mathematics? Because it is one of the liberal arts - the freeing arts. It enlarges the mind, it helps students think in certain kinds of ways that they never would otherwise. Perhaps most importantly, and this comes up most strongly in statistics, mathematics arms students against mathematical propaganda, which is rampant in today's society. The misuse of statistics to convince other people of statements that are, quite simply, false, is something I would want any student to be fully armed against. And without numerical literacy, particularly statistical literacy, this is not possible. So we can see how mathematics frees you from the propaganda of so many out there who want control over your life.
More directly to your question, sure there are various things you can try. Tell the student that they should pretend someone is holding a gun to their temple, and if they make a mistake, you're going to pull the trigger. Or you could always encourage them to work in a high-energy physics lab. With 10,000 volt wires running around, if you make a mistake,
you're dead. Great way to get good at mathematics. For getting good at mathematics unavoidably involves some significant attention to detail, particularly in algebra.
You can also try teaching the Conservation of Symbols Law, formulated by yours truly:
In any derivation, all symbols (digits, decimal points, parentheses, arithmetic signs such as $+, -, \times,=,$ etc.) from one line must survive to the next line unless a specific valid algebraic property is invoked to alter them. Moreover, no new symbols may be introduced in a new line, unless a specific, valid algebraic property is invoked to do so.
I'm not sure I can help more unless you give a few more details, but perhaps something here might help.
