Hello, Thank you for using justanswer. I can assist you with your questions today.
1. No, inherited property is never taxable to the recipient.
2. It really depends on what your mother in law's will says. If it says that your husband inherits the trust then he would keep the trust and the property in it to do what ever he wishes. If the will states that the trust is to be liquidated and the assets distributed then you would not get to keep the trust. I am inlined to say though that you would get to keep the trust as is since your husband is the "successor" as outlined in your questions.
If so, do we pay taxes and at what rate? - if you keep the assets in the trust the trust will generally issue you a Form 1041 Schedule K1 outlining the taxable income allocated to your husband annually. The taxable income "flows through" to your husband and he pays tax on the income at his applicable tax rates, depending on his tax bracket.
The trust will generally keep tax rate character in good form such as long term capital gains, qualified dividends, Section 1231 gain so that he gets the preferential tax rates. Note that long term capital gains are taxed at 15% for most folks and 23.8% for top tax bracket folks. see an article here for some basic information on the tax rates for 2013 - http://taxes.about.com/od/Federal-Income-Taxes/qt/Tax-Rates-For-The-2013-Tax-Year.htm
3. If we transfer a portion of her investments (stocks and bonds) from her trust to a trust in our kids' names, will we owe taxes? - generally not but it depends. You and your husband can each gift up to 13,000 each to an individual like a family member without the gift tax applying. Anything in excess of 13,000 and the gift tax applies, but only if you do not elect to have the gift applied to your lifetime annual exclusion. You would need to file a gift tax return to make the election or to pay the tax. In any event if you gift over 13K you have to file the gift tax return. here is a good link to explain the lifetime exemption - http://wills.about.com/od/termsbeginningwithe/g/exemptgift.htm
click on some of the other links as well as they carry good information about the gift tax which is actually quite complex.
I hope this provides the clarity you were looking for. Please let me know if you have any further questions.