Let's address the first question first. You say that you elected to keep your Family HDHP coverage when your daughter got her own policy on July 1, 2019.
A Family policy is any HDHP policy in which the taxpayer and at least one other person is covered, whether that other person is a spouse or a dependent.
However, the way you phrased this issue makes me wonder if you have a spouse. If the result of your daughter leaving was that you were the only person on the policy, then you should had returned to a Single HDHP policy. If so, you would claim Family coverage for January through June and Single coverage from July to December.
If, on the other hand, you have a spouse or at least one other dependent, then you could retain the Family HDHP policy. If this is the case, then you are permitted to use the annual Family HSA contribution limit for an HSA, which is $7,000 this year (add $1,000 if you are 55 or older, so $8,000 for you).
As for the second question, your daughter began her own HDHP coverage on July 1,2019 if I understood you correctly. She evidently has a Single HDHP policy. If she still had HDHP coverage on December 1, 2019, then the "last-month rule" allows her to use the full annual HSA contribution limit for a Single policy, which is $3,500 in tax year 2019.
The only catch to the last-month rule is that she has to stay under HDHP coverage for the "testing period", which is all of 2020. If she loses HDHP coverage in 2020, then she will report that in early 2021 when she does her 2020 tax return.
Note that this assumes that your daughter has her own HSA, created on July 1, 2019 or shortly thereafter.
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