My husband's employer does not take out taxes...?

If taxes aren't taken out of your pay, you'll have to set up a tax savings account where you deposit some money every payday in order to pay them at the end of the tax year. You should set aside about 20 percent of each payday for that purpose. A lot of side jobs, and contract jobs are like this--and you have to be responsible for paying your taxes yourself.  If you are not using the money you've set aside, then you can use it for something else if you want after you file. 

Should you file separately?  Maybe--you'll have to fill out the return both ways and see if you still get a refund by claiming his wages with yours. Only ONE of you can take the deductions for dependents and any others that have to do with dependents, so it may or may not be a better deal for you to file as married/separate than married/jointly. You can try it and see. 

Does your husband work independently or does he work at a set schedule at the employer's premises for a set hourly or weekly rate?  If your husband is being treated like an employee on the job he should be treated like on on payday.  Contact your state unemployment tax office and ask them to investigate if they think he should have taxes withheld and/or file a Form SS-8 with IRS to get a binding determination.

No. Married filing joint is almost always more beneficial. But for now, you (as a couple) should have been making quarterly payments and or have more withheld from your pay. H needs to go over his withholding needs with employer.

It sounds like he is employed as an independent contractor.  You will have to consult a CPA about how to file yours and his taxes.

No.  He will probably owe more filing separately than we would owe filing with you.

Do they take out no taxes?  No SS, Medicare?

He needs to bring that to their attention, if he is really an "employee".

If he is not an employee, he needs to sent the IRS estimated taxes or he will owe taxes when he files.

* It would not make sense to not file jointly over this.  That would likely increase the amount that both of you owe.

If they are taking out SS/Medicare, but not federal income tax.  He may not be earning enough to have that withhold.  You can check his paycheck at PayCheckCity.com

It means that you should be sending in a check for estimated taxes every quarter.

look up

form 1040-ES

Conventional wisdom dictates that the tax rate for a married couple filing jointly is lower than the tax rates of the two individual filing separately.  That you have withholding and he doesn't makes no difference to that conventional wisdom.