Would that mortgage money count as income in any way for either tax or college tuition assistance purposes? Any other penalties or consequences that could bite us?

The answer on college tuition assistance is: it depends on whether you're applying to FAFSA schools or Profile schools (two different financial aid application forms). For FAFSA schools, the equity in your house is not reported as an asset. This is to your benefit when calculating financial need. If you refinance and pull a lot of money out of your house, it will not count as income for FAFSA. However, the money must be reported as an asset regardless of where you park it. If it's in cash, it will be "assessed" for FAFSA at a rate of 5.6%. So if you have $200K in cash (for example), then $11,200 of it will be included in your expected family contribution to college. This cash plus your other savings might put you out of the running for any need-based aid. (You also have an asset protection allowance so the assessed value will be a little lower).

For schools that require you to fill out the Profile form, your home equity will be required to be reported. How each individual school uses this number is up to its own formula, so it's impossible to say if it will decrease your aid. If you pull out equity as cash, your total assets remain the same (equity is the value of the house minus what you owe on it), and so it doesn't necessarily matter for Profile where that asset is.

Second homes are considered assets for both FAFSA and Profile and their net value reported on both forms.

Here is a link that might be useful: Good info here

I believe you have 5 years to sell your current home to avoid capital gains.

Unless the law has changed recently, you have to have lived in it for 2 of the last 5 years before you sell it. So if you move out and don't move back in, you have 3 years after you move out to sell it and avoid capital gains taxes.