δαί is used in colloquial Attic after interrogative words to express wonder, indignation, etc. Thus, τί δαί; πῶς δαί; what then? how so?
δῆθεν truly, forsooth, is commonly used of apparent or pretended truth, and mostly with an ironical tone. Thus, ἐκερτόμησας δῆθεν ὡς παῖδ᾽ ὄντα με
thou hast mocked me forsooth as though I were a child Pr. 986.
δήπου probably, I presume, I should hope, doubtless, you will admit, is stronger than πού perhaps, I suppose. δήπου often has a touch of irony or doubt in stating a case that would seem to be certain; as ἴστε δήπου ὅθεν ἥλιος ἀνίσχει you know, I presume, where the sun rises A. 5.7.6. In questions δήπου expects the answer yes. οὐ δήπου certainly not and is it not so? (with irony).
δῆτα assuredly, really, in truth, is rare outside of Attic. It occurs: (a) In answers, often when a word is repeated with assent; as γιγνώσκεθ᾽ ῡ̔μεῖς ἥτις ἔσθ᾽ ἥδ᾽ ἡ γυνή; γιγνώσκομεν δῆτα
do you know who this woman is? Yes indeed we do Thesm. 606; οὐ δῆτα surely not, in strong or indignant denial. (b) In questions, to mark an inference or consequence, as πῶς δῆτα; how in truth? τί δῆτα; what then? καὶ δῆτα ἐτόλμας; and didst thou really dare? Ant. 449. (c) In wishes and deprecations (stronger than δή), as σκόπει δῆτα
only look G. 452b, μὴ δῆτα, θῡμέ, μή σύ γ᾽ ἐργάσῃ τάδε no indeed, my heart, do not this deed Med. 1056.