For Medicaid purposes, MAGI is adjusted gross income (AGI) plus these, if any: 

untaxed foreign income, non-taxable Social Security benefits, and tax-exempt interest.

A contribution to your Traditional IRA (same as Deductible IRA) does reduce your Medicaid MAGI. Contributions to a Traditional 401k also reduce your Medicaid MAGI.

See: https://www.healthcare.gov/income-and-household-information/income/

The linked answer does not provide the method for calculating Medicaid MAGI.  A deductible IRA contribution can indeed reduce MAGI for Medicaid purposes.

Is there a difference between a Traditional IRA and a Deductible IRA?

Also, looking forward to 2017, do contributions to a 401K lower income for MAGI?

Traditional IRA contributions can be deductible or nondeductible depending on MAGI for determining deductibility (the MAGI referred to in the answer above) of the contribution.  However, since you are asking about Medicaid MAGI, it seems that your MAGI for determining the deductibility of a traditional IRA contribution would certainly be low enough to allow a full deduction for a traditional IRA contribution.  Of course, to be able to make any traditional IRA contribution you must have sufficient eligible compensation to support the contribution.

Yes, elective deferrals to a 401(k) would reduce your W-2 taxable income (or would provide you with a self-employed retirement deduction) that would reduce your AGI and therefore reduce your both your Medicaid MAGI and your traditional IRA deductibility MAGI.

Do business expenses (sole prop) reduce MAGI?  Do dividends that are reinvested, but not in a 401K/IRA/Roth, count toward MAGI?

Business expenses (sole prop) reduce AGI (which also reduces MAG)I. Dividends that are reinvested, but not in a 401K/IRA/Roth, do not reduce MAGI.