Can I make a $2,000 contributions to both my (non-deductible) traditional IRA and my Roth IRA?

No, you are limited to $2,000 total contributions to traditonal IRA and Roth IRA. You can divide it up whichever way you want (e.g., $2000 to traditional and none to ROTH, or none to the traditional and $2000 to the Roth, or $1000 to each, just as long as the total does not exceed $2000 for any given tax year).

Roll-overs and conversions don't count towards the $2000 limit.