How is "æ" supposed to be pronounced?

You have to distinguish English vowels from English orthography. There are between twelve and fifteen distinct vowels in English, depending on your dialect, but there are only 5 vowel letters in the orthography. This causes no end of problems.

The letter æ was used in Old English to represent the vowel that's pronounced in Modern English ash, fan, happy, and last: /æ/. Mostly we now spell that vowel with the letter a, because of the Great Vowel Shift.

When æ appears in writing Modern English, it's meant to be a typographic variant of ae, and is pronounced the same as that sequence of vowel letters would be. So Encyclopaedia or Encyclopædia, no difference.

English orthography is rule based...except it's not very good at following the rules. Sometimes it uses a regular literal one-to-one pronunciation, at oher times the spelling got stuck centuries ago but sounds changes occurred in speech, and sometimes, the word is written as from the foreign language it was borrowed from but the impossible or unlikely pronunciation is adapted to English mouths and ears.

The pair 'ae' or the single mushed together symbol 'æ', is not pronounced as two separate vowels. It comes (almost always) from a borrowing from Latin. In the original Latin it is pronounced as /ai/ (in IPA) or to rhyme with the word 'eye'. But, for whatever reason, it is usually pronounced as '/iy/' or "ee". Encyclodpeeedia, alumneee (for many female 'alumnae'). Another variant is /ɛ/ in an-eh-sthetic for 'anaesthetic'. Note that many of these spellings are now variants and the more common spelling removes the strange looking 'a'.

Another pair borrowed from Latin is 'oe' is in (the old fashioned spelling) 'oesophagus' where it is pronounced /ɛ/ 'eh' eh-sah-fuh-gus.

You may be mixing up the IPA pronunciation symbol æ and the alphabetic letter æ.

In English text, the letter is used as a slightly old-fashioned form of the Latin digraph ae (also in Latin-mediated Greek words) and in some names from Danish, Norwegian, Old English and a few other languages that use the letter natively.

The pronunciation doesn't have to be anything like the IPA [æ].

For Latin loanwords in ordinary English text, it's essentially equivalent to the letter "e" (so always "encyclopEEdia", "julius cEEsar") but in the study of Latin language and culture it's common to pronounce names and terms in ways more similar to how the original speakers did.

In Danish (etc.) names, you'd adapt to what ever approach you would otherwise use for those names in English text. It may end up actually sounding like the IPA [æ].

In most cases, "ae" or "oe" will result in a long or short "e" sound. These spellings originated in Greek and found their way into English. Many of them have changed as spelling is "reformed," but others have not.

Examples:

• Oedipus - "oed" = "ed"
• oesophagus - usually spelled "esophagus" now
• Aegypt - now spelled "Egypt"
• anaesthetic - sometimes spelled "anesthetic"
• paedophile - now spelled "pedophile"

As for "daemon" -- despite what you will hear from some computer people, it is pronounced "demon" -- and despite what you will hear from some others, they are really only variant spellings. The older spelling "daemon" came to be used in the computer sense, similar to when the "compact disc" was introduced to an international English-speaking audience, the original "disc" was used, even though the spelling of "disc" had mostly been reformed to "disk" by that time. This resulted in the current situation in which "compact disc" and "hard disk" are spelled differently.

Now... When "ae" is used at the end of a Latin word, it is technically pronounced "eye." I say "technically" because it's confusing that the "real" pronunciation of "alumnae" sounds like the the popular pronunciation of "alumni" (which "really" should be pronounced "ah-loom-nee" which probably only happens inside a Latin classroom.