If you're seeking a way to verify a deity then you shouldn't go into practicing magic because if you seek something that's not out there or can't be explained what are you going to do with magic? It's a belief that falls into a religion (Wicca.).
Firstly, practice your path however you like, be that with a scientific approach, a spiritual approach, or a bit of both (as I do.) It's your life, your craft, and your practice.
However, in terms of deities there is no broad, generalisable evidence, but instead you must seek out evidence of the Gods in your own life. When you attune yourself, and become aware of the ways in which each of the Gods present themselves to us, you'll find evidence of Them in a form possibly quite specific to you and your relationship with them.
These are my views and opinions on the subjects raised. Good luck :)
well she doesn't work for free, girl's gotta get paid lol. kind of ties in with your first question, if you believe in her, yes. you should introduce yourself to her and ask to learn from her. you should recieve and answer within the month [and if not, thank her and move on for a while. sometimes you're not ready or unable to hear a deities response. work on meditation and mindfulness. chakras might also help you] if you get a yes [keep an open mind, you probably won't see her before you and she tells you 'yes' you might get a dream, a vision while meditating, have a dog follow you. personally i was eating lunch and i got a strong feeling of my Goddesses presence after 3 days of trying to connect with her. it really depends] now that you have met Hecate, you can do many things, bless your meal in her honour, set up an altar and leave offerings, pray to her in circle. even little things like observing the moon cycle, or living your life in her image [like asking yourself 'what would Hecate do in this situation'] as you continue you'll grow closer and better understand each other.
3 give a reason that tells Hecate is real
i don't work with Hecate, so i can't prove her existence, but she's got many followers, does that count? regarding my Goddess, i know she's real because i felt an instant connection when i discovered her, and have felt her presence stronger than other deities i've worked with. i also know she's real because i get rewards/punishments the day or day after i do something. like i asked her for help finding a job and i got a call the next day, i gave her a less than favourable offering and my journalling pen [which i love] went missing, and stayed missing for almost a week until i apologised, and then, randomly, it was in my journal the same day i said sorry. there's many other ways, but it's a feeling and a personal experience, so it's hard to say yes or no without a shadow of a doubt, you need to take a leap of faith really.