In the spoken language, these examples are strings of words where the realisation of the strings in speech is quite different from the citation forms of the individual words. A citation form is the phonetic form of the word when we mention the word without using it in its normal sense. So for example we might say:
Here the item can isn't being used in the same way that it is when we say She can dance for example. The citation forms of function words (the kind of words that concern grammar more than vocabulary) often differ from the forms we hear in normal speech. One reason for this is that they are stressed when we cite them and often aren't when we actually use them.
So in the first example, for instance, we see kinda which is an orthographic rendering of the string /kaɪnd ə/. Here you will notice that the word of is represented by /ə/ and not by /ɒv/, the citation form of the word. In the example betcha we see that /j/, the sound represented in writing by the letter Y appears to have changed into a /tʃ/ so that we have the form /betʃə/ instead of /bet ju:/. The string /bet ju:/ is what we would expect if we added the citation forms of bet and you together.
The orthographic items in the Original Question illustrate two aspects of English phonetics and phonology. The first is the occurrence of WEAK FORMS in the language. The second is the existence of FAST SPEECH RULES or CONNECTED SPEECH PROCESSES, also sometimes referred to as PHONOSTYLISTIC RULES.
English is a 'stress-timed' language. What this means is that the syllables in English utterances do not come at regular intervals in the way that they do in Japanese for example, or in Spanish. Instead English utterances give the impression that the stressed syllables come at regular intervals. In actual fact this is not strictly what's happening, in the sense that although they give this impression, the stressed syllables do not occur at strictly regular intervals at all.
The effect of this is that words that don't carry stress in English are much less prominent than stressed ones. Now, all the material coming in between the stressed syllables of English utterances needs to be said more rapidly so that the stressed syllables don't get pushed apart. We don't want to spoil the stress-timed effect. One of the mechanisms that English has for achieving this is that finicky grammar words - auxiliary verbs, pronouns, prepositions, infinitival-to and so forth - tend to have two forms. They have a so-called strong form when they are stressed. This is the same as the citation form. And they also have a weak form, one with a reduced vowel, normally a schwa, /ə/, that we use when they aren't stressed (strictly speaking, there may be several weak forms of a single word). So the citation form of can, for example, is /kæn/ and the weak form is /kn/ or /kən/. The weak form of to is /tə/.
The realisation of the weak forms of these words can be subject to complex rules and depend, for example, on whether an item is utterance initial or not. So, to illustrate, the weak form of the word he in Southern Standard British English is /hi/ when utterance initial and /i/ when not. In slow careful speech, the weak form of of in English is /əv/. However, in rapid or relaxed speech it may be realised by either /v/ or /ə/. So the string man of may be realised as:
Interestingly, the weak forms of the auxiliary verb have may be realised in the same ways when not sentence initial. So weak have and of are often homophonous in English (hence people's occasionally writing could of been there instead of could have been there and so forth).
So in rapid speech the strings kind of, sort of, lot of may all occur with the word of represented just by a schwa, giving us: /kaɪnd ə/, /sɔ:t ə/ or /lɒt ə/. The strings could have and should have may be similarly be realised as /kʊd ə/ or /ʃʊd ə/
Is there a name for such strings relating to spoken English? I don't think so (of course an absence of evidence is not evidence of absence, there might be one that I haven't come across). The reason is that they are all just examples of a word followed by a weak form of another word. They are two-word strings. Their unifying feature is the occurrence of weak forms, in particular those found in rapid or relaxed speech.
Connected speech rules
Connected speech rules often concern the elision, assimilation and coalescence of sounds when words are used in actual utterances. These are often dependent on the environment in which these sounds occur.
One example of such a transformation is what's known as COALESCENT ASSIMILATION. The adjustments we make to be able to pass smoothly from one sound to another sometimes result in a third different sound taking the place of the two original sounds. This happens very often in English when some consonant is followed by a /j/, the sound at the beginning of the word yoyo. It is often referred to as YOD COALESCENCE. Relevant to us right now is the fact that a sequence of /t/ and /j/ often results in a new sound /tʃ/ (the first sound in the word chair) which replaces the original two segments. So the sequence last year may be realised as /la:stʃɪə/, "las cheer". [SSBE transcription]
In rapid speech the sequence bet you is likely to be realised as /betʃə/ (or /betʃu/) where the /t/ from bet and the /j/ from you are subject to coalescent assimilation. The weak form of the word you is often realised as /jə/. The string /betʃə/ therefore involves both a weak form and coalescent assimilation.
The orthographic items oughta and lemme can also be analysed using connected speech processes in conjunction with weak forms. To cut a long story short, with regard to oughta, for example, the weak form of to is /tə/. Along with a process of degemination (the reduction of a cluster of two identical consonants into a single length consonant), this gives us the form /ɔ:tə/.
So what then?
It seems that the orthographic items listed in the original question have something in common, which is that they exhibit features of English seen in connected speech, and to varying degrees more often in relaxed or rapid speech. The realisations of the orthographic items oughta, betcha and lemme display the feature of two words sharing a single phoneme. The /t/ in oughta belongs both to the word ought and the word to. The same also goes for the /tʃ/ in betcha and arguably the /m/ in lemme. There may well be a word for pairs of words that share a consonant like this (it isn't quite the same thing as liaison). Unfortunately, I don't know it. However, this phenomenon doesn't seem to be something special to do with these items in particular. It's happening for different reasons in each case in our examples.
In terms of speech these three examples don't seem to share all that much with the examples with have or of. The latter just seem to be sequences of two words, despite the orthography.
One thing we might say is that in careful, slow speech some of these sequences will not be a feature of many speakers of Southern Standard British English or General American, even if they do occur in their rapid speech, or relaxed speech. They could therefore be umbrellaed under what has been described (and referred to here in another answer) as relaxed pronunciation.
Here is the Wikipedia introduction to relaxed pronunciation:
Relaxed pronunciation (also called condensed pronunciation or word slurs) is a phenomenon that happens when the syllables of common words are slurred together. It is almost always present in normal speech, in all natural languages but not in some constructed languages, such as Loglan or Lojban, which are designed so that all words are parsable.
Some shortened forms of words and phrases, such as contractions or weak forms can be considered to derive from relaxed pronunciations, but a phrase with a relaxed pronunciation is not the same as a contraction. In English, where contractions are common, they are considered part of the standard language and accordingly used in many contexts (except on very formal speech or in formal/legal writing); however, relaxed pronunciation is markedly informal in register. This is also sometimes reflected in writing: contractions have a standard written form, but relaxed pronunciations may not, outside of eye dialect.
I am not personally familiar with relaxed pronunciation as a technical term. And I don't vouch for Wikipedia as a source. However, some of the editors and commentators on that specific Wikipedia page are professional phoneticians. In particular Peter Roach, Emeritus Professor of Phonetics at the University of Reading, and author of English Phonetics and Phonology has commented on the page offering constructive criticism. I therefore suggest that these are all examples of RELAXED PRONUNCIATION. As the Original Poster points out, in term of speech as opposed to orthography, they aren't contractions.