Technically any size will effect the tides but I'm assuming you mean a noticeable effect.

It also depends on what you mean by effect the tides. For instance if an object 1/10 the mass of the moon were to suddenly appear it would be "in competition" with the moon as it were. So although it wouldn't have an effect on the tides in and of itself it would reduce the effect the moon had on the tides by 1/10.

If you wanted the tides to be caused by the new object than it would have to have a mass greater than the moon and significantly so for the effect to be seen since again the moon would still be effecting the tides.

I don't understand what you mean about gravity, do you mean pull the earth out of it's orbit of the sun, or would this hypothetical new object cause the earth to orbit it. Essentially Turing the earth into a moon of this new astronomical body?