I am wondering about the long vowel e, and whether it is -ee or -ei. Let's use hē as an example, is hē written as へい or へえ?
Long vowel E is usually written using い, so, へい is correct. There are some exceptions where it is written with え such as ねえさん、but these are exceptions to the rule, and don't change pronunciation.
There are also a few other ways the vowel can be written as extended: sometimes the ー mark will be used just as in katakana, and sometimes a smaller え is used, for example when ending a sentence with ねぇ. However, again, these are exceptions, so it's best to assume that the vowel is extended with い.
In theory, “ええ” and “えい” are two distinct phonemic segments, mapping to /ee/ and /ei/ respectively.
In practice, however, the latter is often pronounced as the former, even in increasingly formal contexts, but theoretically they are distinct. — we are currently witnessing the middle of a phonemic merger in a language. As time goes on, one would expect the /ei/ pronunciation to sound increasingly more outdated, until it is considered completely incorrect, and the segments have completely merged.
Thus, “えい” can always be pronounced as “ええ”, but not vice versā.
This rule does not apply across morpheme boundaries; across morpheme boundaries, such as in the common “している”, it must always be pronounced /ei/.
Thus “ええ” is the correct “long e”.
It should also be said that phonemically Japanese does not have “long vowels” which are a phonetic realization if successive identical vowels, or, in some analyses a vowel an a chroneme. The underlying form of [e:] can either be analysed as /ee/ or /eR/ depending on whom one speaks to, never as /e:/, which is applicable to, say, Latin, which has a true distinction between short and long vowels.
This discussion provides some insight from some native speakers on the matter.
The proper way to write the extended え-sound (hē, in your question) would usually be written as へい. In casual writings though most people don't make such a distinction.
Read the explanation for long sound here.