Do you think Padfoot’s personality influences Sirius? If so, how do you think it manifests?
A lot of Sirius's key traits could definitely be interpreted as being very "canine".
His strong sense of loyalty and trust. Is there any more stereotypical description of a dog than as a loyal, trusty companion? Sirius is intensely loyal and devoted to those lucky enough to be appreciated by him. These traits may have contributed to his downfall (his misplaced trust and failure to appreciate that not everyone values loyalty as strongly as he does) but they are big parts of what makes him a good person at heart, and I definitely think they play a large part in why his animagus is a dog.
On the other hand, one very canine trait that Sirius displays is not always a good thing. Canines are pack animals - they live in family groups. They typically do not feel comfortable when separated from their family. It can make them feel unsafe, insecure, and foster negative behaviours which many people see as a result of their pets' separation anxiety.
We could theorise how Sirius's separation and ostracization from his blood family affected his behaviour (I personally believe that the Snape incident occurred not long after he ran away and that this moment of recklessness was largely fuelled by his emotional hurt and insecurity stemming from his family troubles), but the main canon example of Sirius's troubles with being separated from whom he views as family is James. He and James were intensely close, more so after Sirius ran away from home, I'm sure. And his behaviour post-Azkaban largely stems from his un-processed feelings surrounding James's loss. He's had no time to come to terms with this world without James, and with his lack of family, and this unsettled lack of belonging contributes to his reckless behaviour that we see in OOTP. Also, I can't remember if there's any canon basis for this, but I definitely headcanon that he struggled with James's relationship with Lily at first. Someone new mustling in on their own closeness and accepting that he is no longer the most important person in James's life must have been A Process. Many people see similar situations arise, with their dogs struggling to accept their owners' new partners, so I definitely feel this canine trait influenced Sirius's impressions of Lily prior to her being accepted into his trusted circle of pack members who would benefit from his unwavering loyalty.
As a side-note, I was very much drawing on Sirius's Animagus form when I described Walburga feeling that he needed a task to focus on in Circinus. Her reminiscing about how he would become restless and naughty as a child when he had nothing stimulating to do was very much me trying to compare him to a puppy chewing up the furniture when given nothing more challenging to do, lol.