OP: Have we done anything wrong?
When we look at whether or not someone can contribute to an HSA, we need to determine whether or not that person is an eligible individual. The definition of an eligible individual is found in IRS Publication 969. It says:
To be an eligible individual and qualify for an HSA, you must meet the following requirements.
You are covered under a high deductible health plan (HDHP), described later, on the first day of the month.
You have no other health coverage except what is permitted under Other health coverage , later.
You aren’t enrolled in Medicare.
You can’t be claimed as a dependent on someone else's 2017 tax return.
This means that your girlfriend is an eligible individual, because her only health coverage is the HSA-eligible HDHP, but you are not an eligible individual, because you have other non-HSA-eligible health coverage. The HSA you are talking about belongs to your girlfriend and not you, so no problem there; your girlfriend is eligible to contribute to her own HSA.
The next thing we have to look at is to determine what your girlfriend's maximum contribution limit for the year is. In 2018, if you have self-only HDHP coverage for the year, your contribution limit is $3450. If you have family HDHP coverage for the year, your contribution limit is $6900. It is important to note that for HSA purposes, family coverage is defined a little differently than in other contexts. Here is what IRS Publication 969 has to say about it:
Self-only HDHP coverage is an HDHP covering only an eligible individual. Family HDHP coverage is an HDHP covering an eligible individual and at least one other individual (whether or not that individual is an eligible individual).
What this means for you is that because your girlfriend's HDHP plan covers both her and another person (you), she is considered to have family HDHP coverage and can contribute at the full family coverage level. As long as the plan covered both of you for the full 12 months of 2018, her contribution limit is $6900. It doesn't matter that you are not an eligible individual because you have other coverage.
However, you should know that although your girlfriend is allowed to contribute to her HSA at the maximum family limit because you are on her plan, she is not allowed to pay for your medical expenses out of her HSA. IRS Pub 969 says:
Qualified medical expenses are those incurred by the following persons.
Since you are not her spouse or her dependent, she cannot pay for your medical expenses from her HSA.
OP: My question is whether we are still able to file single separately for 2018.
Yes, if you are not married, you each need to file separate tax returns as single. Your girlfriend will have HSA contributions to report; you will not.
OP: Now that open enrollment is around the corner, should I remove myself from her HSA plan? Or is it kosher for me to be there so that we can take advantage of maximizing her contribution?
If your goal is to allow her to maximize her contribution, then yes, you can stay on the plan if your girlfriend's employer allows it and if it is not costing your girlfriend more in premiums than it saves her in taxes.