In connected speech, /ð/ at the start of function words may be assimilated to a preceding consonant in some cases. However, I don't think there are any circumstances where this kind of assimilation always occurs—my impression is that it is gradient. Also, the identity of the preceding consonant probably affects the probability of assimilation.
I have found a source "Applied English Phonology", by Mehmet Yavas, that gives a more specific description of the conditions of this assimilation:
unstressed initial /ð/ in words such as the, this, that becomes assimilated (with or without complete assimilation) to previous alveolar consonants (e.g. what the heck [wɑt̪d̪əhɛk], run the course [ɹ̣ʌnːəkɔɹ̣s], till they see [tɪlːesi], how's the dog? [haʊzːədɔg], takes them [teksːəm]) (p. 67)
I think "takes them" may not be the best example of the phonetic process in question, since them additionally has an alternative form ’em that may occur after any consonant, not only alveolar consonants.
The fact that "in the" could be realized as [ɪnːə] or [ɪnə] rather than [ɪnðə] is mentioned in Geoff Lindsey's blog post "Lucas quiz – the answers".
Another known phonetic phenomenon is deletion (which could be seen as assimilation followed by mandatory shortening) of [θ] or [ð] before the suffix -(e)s. This is lexicalized for many speakers in the noun clothes /kloʊz/, although the non-assimilated pronunciation /kloʊðz/ is not uncommon either. Some speakers (I think a smaller number) also have this type of assimilation/deletion in the word "months", pronouncing it as [mʌnts]. This has been covered in other places on this site (e.g. How to distinguish 'month' and 'months' in pronunciation?)
As far as I know, no native speakers (without speech impediments) use [z] for /ð/, or [s] for /θ/, in contexts other than assimilation to an adjacent /s/ or /z/.
Some native speakers do use realizations other than [ð] and [θ] more generally—I discuss this in more detail in my answer to Do all native English speakers actually pronounce the "th" sound?—but as far as I know it is always something non-sibilant like [d̪], [d̪͡ð], [v]. If you can't manage [θ] in "thorough" or "thief", I would say it's better to fall back on [f] or [t] than to use [s].
As for "at the beginning" and "what the heck", if you pronounce them at a reasonable pace, it will probably not even be noticed if you use a dental stop [d̪] rather than a dental fricative.