Do I have bottle bombs? Two questions about carbonating.

Others may disagree but I don't know that you are providing enough information. You say you bottled the mead when it was at 1.000 OK but you don't say what the starting gravity was so I cannot say that the yeast still has enough wherewithall to eat their way through this priming sugar. Bottle bombs? Hard to know. Alcohol has a lower gravity than water and pure water has a gravity of 1.000. That means there must be fermentable sugars in the mead you bottled - perhaps 4 points or more. To that you added a little more sugar... is this enough to create a bottle bomb? Well, assuming the starting gravity was not in the stratosphere and the yeast can stil eat their way through all this sugar , no, I don't think that this will result in exploding bottles but it may result in gushers when you open those caps...but if there is no protein net in the mead (not a braggot, for example) then the excess CO2 may not create a volcano when you pop open the cap.

My rule of thumb is that 1 lb of honey mixed in water to make 1 gallon will raise the gravity of the water to 1.035. So, 3 lbs will raise the gravity to 1.105. My second rule of thumb is that you multiply the SG by 131 to get a good estimation of the potential ABV which in this case is a scant 14% . So IMO the yeast is not near its limit. Bottom line: I would expect the yeast to be able to reduce the priming honey to CO2 and ethanol and the yeast may also be able to pull out more CO2 and alcohol from the residual sugars in the honey (though some of the sugars in the honey may pose more of a challenge to the yeast)..

1/4 tsp of honey per liter is not enough for noticeable carbonation. The other possibility is that the mead is still fermenting and 1.000 isn't as low as it will go. It takes 2-3 gravity points to get beer level carbonation, so if the mead finishes at 0.998 in your bottles you will get fizz and no bottle bombs.