Of course you pay for training, why wouldn't you?
Working for competing companies is even stupid than being overinsured, and could potentially be illegal (for the employee).
Company withholdings don't change so this is not your concern.
1. The employee's insurance pays; in the US, they pay at least part of their insurance costs so it is unlikely for them to be covered by two services. If so, they can choose. If they were injured on the job, or in some capacity based on your work, then how is it different? Actually, how is this situation different from them purchasing outside insurance or being covered by their spouse? It's unlikely to be a problem and if it is it's not unique.
2. Sure, why not?
3. What does this have to do with them having multiple jobs? If they steal your intellectual property, that's a problem whether or not they are employed elsewhere. How does double employment compound this? If you hired someone from another job and they brought stolen property with them, is that different?
4. They are responsible for letting you know how much to deduct. You can do standard mandatory deductions based on expected salary and as long as you pay the required employment taxes based on the salary you pay them there isn't a difference in withholding on your end, only on theirs. If they owe more in tax, what do you care?
Ever had a medical claim by your over employed staff? Which insurance pays?
Do you pay for training?
In case of intellectual property theft from you or the other company, do you know what are your obligations? (Receiving company is can held liable.)
What amount of taxes do you withhold? What are the penalties for withholding significantly wrong taxes in your jurisdiction?