Spirits may come in many different ways. They take the form of an animal. They may be a familiar or just another spirit trying t get across from you. The spider seems more like a familiar. Familiars are the spirits that want to help you, and protect you and those you hold dear, on your magical journey. They can form a telepathic link to you and can find you where ever you are. Even if you don't want them, they are extremely loyal and will not leave your side. Keep in mind that familiars aren't seen by others, only you. I say start meditatinbg and open your spiritual side. That spider may be a familiar trying to connect to you.
The horse on the other hand may have been a coincidence. You just might have an empathic link to all of mother earth's animals. Your energy might've connected with the horse's energy, making it curious and attracted to see you.