What is the actual spelling/pronunciation? What is the origination of this phrase?
Supposed to is a very common phrase, as in: you're not supposed to come here. It means "you should not come here", "you are not meant to come here". Using *suppose to in this sentence would be considered wrong by most educated speakers. The cause of this common omission of the d is probably that d t sounds the same as t, so that there is no difference in pronunciation between supposed to and *suppose to.
To suppose something means to assume something. When used in a passive construction, you use a form of the verb to be plus the past participle of to suppose, which is supposed. To be supposed to means to be expected to, to be obliged to. You can see that the verb gets a somewhat different sense in the passive voice from what would normally be expected.
The original meaning of "suppose" is epistemic: "think", or "assume".
But various modal verbs, such as "must", have both deontic ("you must go" = "you ought to go") and epistemic ("you must be" = "I conclude that you are") meanings.
In the passive only, "suppose" has acquired a deontic sense.
Nowadays "suppose" nearly always takes a sentential complement ("I suppose that ... "), but in older or more formal English it can take a direct object and an epistemic infinitive complement "I supposed him to be ... ".
So "He was supposed to go" is the passive of a construction which is now rather restricted, in the epistemic sense of "It was thought that he was going". But in the specialised deontic sense, the construction has survived.
The expression "supposed to", when used to describe what someone should do, is always spelled "supposed to" and always goes with a form of the verb "to be" (E.g. "they are supposed to do X", "I am not supposed to do Y").
Historically, it originated as a passive-voice version of the verb suppose, meaning in this context "expect". The "supposed" part is therefore in origin a past participle, which is why it has the "-ed" suffix.
But in modern English, this expression doesn't really behave like an ordinary passive-voice version of to suppose. For one thing, I believe everyone pronounces it differently: "suppose" is /səˈpoʊz/, ending in voiced /z/, while "supposed to" is /səˈpoʊstə/ or /səˈpoʊstu/ (or, as mentioned in the comments, /ˈspoʊstə/), with voiceless /s/ after the /oʊ/. This kind of phonemic voicing assimilation between words is unpredictable and not common in English; so I think "to be supposed to" should be classified as fixed expression that has become somewhat distinct from the ordinary verb "suppose", just as "to be born" has become somewhat separate from the ordinary verb "to bear".
We see the same devoicing of originally voiced fricatives in certain other verbs that have developed specialized meanings in fixed expressions containing the word "to":
- "have to"/"has to": normally, "have" ends in /v/ and "has" ends in /z/, but "have to" has /f/ and "has to" has /s/. (But "had to" retains a voiced /d/ phoneme, which is not a fricative.) This only applies to "have to" as a fixed expression with a specific, special meaning; it doesn't occur when the word "have" or "has" just happens to come before "to", as in "She will give all she has to fight cancer."
- "used to" meaning "once did" has /s/, even though the ordinary verb "use" has /z/.