If you re-read your post, you will find quite a few

popping up.
When I first started dating this girl, we were looking at her iPad, and she got a text from this guy. I asked her who that was, she said "oh nobody".

She later told me it was a guy she had been talking to, and didn't want to start a fight.
7 months into the relationship, she wanted some space

so we didn't talk for a week and a half. For 5 of those days, she was in the hospital. So we got back together, and I take her out for a night out on her birthday, she confesses that she went out on a date with the guy.

Of course, she only had a couple days to actually date someone.
And now, a year and a half into the relationship, we had another break up. We have since started sleeping together. She says she loves me and I say I love her. We only get together for sex which is about once a week.

However, I have noticed that his name appeared on her snap chat app as a best friend. She swears that it was her 21 yeast old daughter exchanging pictures with his son.

However, snap chat resets after 1 week of not sending pics. So clearly, she is exchanging pictures with this guy.

He has been her top friend on it for about three weeks.

She also swears that she has not seen him in person. I am not with her very much,

so she could easily be seeing this guy. I think it is very evident that she wants something with this guy from her actions.
I am thinking that it is time to cut ties with this one. She straight up lied when she said she never sent any pics to this guy.
From what you describe, it doesn't sound like she suffers from BPD. More likely she's just not ready to settle down and either is afraid to communicate that or is outright lying too you. If you're not ok with continuing on with a casual relationship, then you need to discuss it with her. If she's not ready to be on the same page as you, then you may need to be prepared to walk away.