John Frawley recommends using the Regiomontanus house system for horary questions and points out that William Lilly had much success with it. Since both Lilly and Frawley are highly esteemed experts on horary, I have followed suit.
The querent is Saturn in Aquarius in the second house and the Moon in Libra in the 10th. Small animals are sixth house concerns so a new dog, the quesited, would be ruled by Mercury because Gemini is on the cusp of the sixth house. Mercury is retrograde in Virgo in the 9th house. Her current feisty little dog would also be Mercury but one can’t use the same significator for both dogs.
Also, she wants to know how the two dogs would get on, so we have to be able to see how they relate to each other. We cant do this if they have the same significator.
She is asking about a new dog. We could assign the new dog to the seventh house in the sense that it will be her next dog but that is ruled by the Moon which is one of the querent’s significators. Frawley says we can use another planet (not the house ruler) if it is within a couple of degrees of the cusp of the house in question and is in the same sign as that cusp. Given what we know about her current dog’s personality, we can safely take Mars, the planet in the sixth house to be her current dog.
The current dog is Mars in Gemini so the new dog, Mercury, has rulership over it. The current dog will respect the new dog and the latter may dominate the new dog since Mercury rules the 6th house. The current dog is Mars at 18 degrees of Gemini where it is peregrine. A peregrine planet is debilitated and has no essential dignity.
Will They Get On?
The Mars nature of the current dog may not be happy to be dominated by Mercury. A ruler is not always benevolent and can be someone to be feared. Mercury is clever but volatile and the fact that he is retrograde may mean that the querent might have to regret her decision about getting another dog in one way or another if she gets one now.
How will the new dog receive the current dog? Well, Mercury is both in its rulership and exaltation in Virgo in the ninth house. It is the planet with the most essential dignity in this chart and is very powerful.
We have to see how Mercury in Virgo receives Mars. Consultation of the Dorothean Table of Essential Dignities shows that Mercury in Virgo receives Mars as peregrine. The new dog will be indifferent to the current dog. This makes it at best unlikely that they will be friends.
More concerning is that Mercury is conjunct the fixed star Benetnash, also called Alkaid. It is of the nature of Mars which does not augur well for a relationship with the current dog. The star is associated with violent death, mourning and ruin though pride and folly. When conjunct Mercury, it can signify deafness. This could mean that the new dog will be physically deaf but it could also mean that it will be disobedient and won’t listen to her.
Mercury’s antiscion is at 2 degrees of Aries, further emphasising the hidden Mars nature of the new dog. The current dog, Mars in Libra, has its antiscion in Cancer ruled by the Moon and so has a gentler side.
Since the querent is the Moon in Libra, she cares a lot about being fair to her current dog. Mars is also trine Saturn, so is in a happy relationship with the querent. The querent is probably also thinking about the expense of a new dog because her significator, Saturn, is in her second house. Incidentally, Saturn in Aquarius receives Mercury, the new dog, as peregrine so the new dog is not likely to make her happy either.
All things considered, it is probably not a good idea for the querent to get another dog. Her current dog is unlikely to be happy to be dominated and Alkaid is a nasty star. There is also a possibility of some very unpleasant and violent dog fights in which her current dog may be seriously hurt.