Syllables with secondary stress are not unstressed
First of all, it's important to recognize that (aside from certain exceptional cases for some speakers*), /ə/ cannot occur in any kind of stressed syllable: not in a primary-stressed syllable, but also not in a secondary-stressed (or "tertiary-stressed") syllable either. The vowel phoneme /ə/ only occurs in syllables without any kind of stress.
The first syllable of anesthesia has secondary stress, as shown in your transcription /ˌæn.əs.ˈθi.ʒə/. The existence of this secondary stress is also fairly predictable from a certain rule: all English words have some kind of stress (either primary or secondary) on at least one of the first two syllables. And when the third syllable of a word is stressed, a secondary stress usually falls on the first syllable. That explanation accounts for /ˌæksəˈdɛnt(ə)li/, /ˌæljuˈmɪn.j.əm/, and /ˌsætɪsˈfækʃən/ (the last should be transcribed with secondary stress on the first syllable). The vowels in the first syllable of these words are not "in an unstressed position" for the purposes of vowel reduction.
When the form aluminum is used, the second syllable is stressed, so the first syllable is unstressed and reduced to schwa: /əˈlumɪnəm/. It's the same for /ə/ˈparent and /ə/ˈlumni. In phoˈtogr/ə/phy the second syllable is stressed, and the third syllable is fully unstressed.
Fully unstressed closed syllables can have /æ/ or /ə/
It remains to discuss /mæɡˈnɪfəsənt/ and /ˈæk.ɹo.bæt/. In both of these, the /æ/ is in a "closed" syllable (a syllable that ends in a consonant). Unfortunately, it is possible for /ə/ to occur in a closed syllable, so you can't completely predict the use of /æ/ rather than /ə/ in these two words. There are some patterns in terms of the pronunciation of similar words; e.g. you also find /æt/ at the end of aristocrat and democrat.
There are some differences even among native English speakers in the use of /æ/ vs. /ə/ in fully unstressed syllables: for example, in this blog post by the British phonetician John Wells ("strong and weak"), you can see the word gymnast transcribed as ˈdʒɪmnæst, whereas I (an American English speaker) would transcribe the vowel I use in the last syllable as /ə/. So using /ə/ more often than native English speakers won't necessarily make what you're saying incomprehensible. I would guess that there are some words where it might cause confusion, though (although I can't think of a specific example right now).
In some theories, syllables with unreduced vowels have "tertiary stress"
Although I referred to "fully unstressed" syllables in the previous header, there are some phonologists that argue that the presence of an unreduced vowel like /æ/ actually implies that the syllable has some kind of stress. That's the concept of "tertiary stress" that I mentioned in passing at the start of this post. In some—particularly American—transcriptions, you might see tertiary stress transcribed with a lowered stress mark (the same symbol used for secondary stress), so the word acrobat might be transcribed as /ˈæk.ɹoˌbæt/. Although the American Heritage Dictionary doesn't use IPA transcriptions, you can see that it uses its own style of stress mark on the last syllable of the word acrobat.
This is just an issue of notation and analysis. The presence of "tertiary stress" is unfortunately not as predictable as the presence of secondary stress, so I can't give a simple rule for it.
Further notes on the position of secondary stress
I said that you can usually assume that a word stressed on the third syllable has secondary stress on the first, but there are some possible exceptions to that rule of thumb. For example, electricity can be pronounced with secondary stress on the second syllable and no stress on the first (eˌlecˈtricity; but an alternative pronunciation ˌe.lecˈtricity also exists). Another way of determining secondary stress is by reference to related shorter words. For example, accidentally is related to the shorter word accident: the primary stress in ˈaccident corresponds to the secondary stress in ˌacciˈdentally. Likewise, ˌsatisˈfaction is related to ˈsatisfy. But that method only works when there is a related shorter word, and you know the position of the primary stress in that word.
*It's not important to memorize, but the main example I know of for a word that may have stressed /ə/ for some speakers is just in the sense of "only", contrasting (for these speakers) with the /ʌ/ used in the adjective just. I don't think this distinction is common, though: many speakers have the regular pattern of using /ʌ/ in just whenever it is stressed, and /ə/ only when it is unstressed.