Auxiliary (or Helping) Verbs

Forms

English has the following auxiliary (or helping) verbs:

Forms of be: be, am, is, are, was, were, been, being

Forms of do: do, does, did

Forms of have: have, has, had, having

Modals: can, could, may, might, must, ought, shall, should, will, would

German has the following auxiliary (or helping) verbs:

Forms of sein: sein, bin, bist, ist, sind, seid, war, warst, waren, wart, wäre, wärest, wären, wäret, sei, seist, seien, seiet

Forms of haben: haben, habe, hast, hat, habt, hatte, hattest, hatten, hattet, hätte, hättest, hätten, hättet

Forms of werden: werden, werde, wirst, wird, werdet, wurde, wurdest, wurden, wurdet, würde, würdest, würden, würdet

Modals: dürfen, können, mögen, müssen, sollen, wollen

How are auxiliary verbs in English the same as, or different from, auxiliary verbs in German?

German never uses a form of do as an auxiliary. Forms of do are used only for the main verb. Examples: Ich tue das nie. (I never do that.) Ich mache meine Hausaufgaben. (I do [am doing] my homework.)
German never uses a form of sein to express a progressive tense (am going [gehe], was singing [sang, habe gesungen])
German uses werden instead of sein to form the passive voice.
German uses modals as auxiliaries combined with infinitives.