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.
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.