Sorry to hear of the loss of your grandparents.
As the assets do not require probate court and have transferred to my mother by virtue of the account titles, is this still considered an "estate"?
Yes, all of the assets whether or not going into probate are part of the estate tax and for income tax purposes. For example, one could not avoid estate tax just by having accounts worth ten million dollars pass to their children by titling.
Of course, once those assets (immediately) pass then the income from those assets is no longer income of the estate but of the recipient.
...am I correct in assuming that my mother will not need to file an estate tax on behalf of the estate?
Yes, that is correct when the value is less than the amount for required filing.
...because the money has passed into my mother's name without probate does it become a "gift" situation with my mother gifting the money to her siblings?
Because your mother got the funds she has inherited those funds.
If she gifts more than 14,000 to any one person in a year she has to file a gift tax return.
Since she has a five million (and a bit more) dollars to give in gifts before she will owe any tax there is just a Form 706 and zero tax to pay with that return required. The amount gifted does reduce her amount that will pass free of estate tax on her departure.
For more details for her filing see http://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Filing-Estate-and-Gift-Tax-Returns
Her siblings are getting gifts from her and not an inheritance but neither gifts or inheritance received is subject to income or gift tax to the recipient. Income earned on the gift after receipt would be included in the recipient's tax return.
Please ask if you need more discussion or clarification.