Structuring extensive medical histories + demographic information for prediciting future medical outcomes
There are many options and many different problems here, I think you should start simple and then try to improve on it.
Imho the most basic option is to restructure the dataset so that there's one instance for every patient at any given time $t$, for instance every month. The binary label represents whether the patient was admitted to hospital in the period $t$ to $t$+60 days. The features should represent whatever relevant indicators can help the model based on past data (before $t$), for instance mean/standard deviation of the blood pressure in the past N months, possibly with various values of $N$ and/or a sliding window. This way both short-term and long-term information can be integrated. Occasionally you might have missing values, it's important to take this into account.
In this design there will be several instances for every patient, and this could cause a bias in the evaluation. There are two options: either there is one independent model for every patient, but this is probably too complex. Or the datasets is made of all the patients, and in this case it's important to avoid having the same patient in the training set and test set.
The problem of patients who would take an ambulance to the ER on a regular basis is a preprocessing problem: if possible, these instances should either be discarded or re-labelled as if it was a regular consultation, for example.