No, you cannot omit these sounds in general. English does simplify these kind of clusters sometimes, but only specific ways are acceptable. The simplifications are generally "assimilations." I found some description of this in this downloadable document by Rachael-Anne Knight: "Understanding English Variation."
Basically, /d/ may assimilate in place to a following stop. So it might turn to /b/ before /b/ or /p/, or turn to /g/ before /g/ or /k/. The /d/ is pronounced more like /b/ or /g/, but it is not dropped entirely: there are still two consonants in the middle of the phrase, and two consonants have a longer duration than a single consonant. (Note: an assimilated /d/ may actually still be distinguishable in some way from original /g/ or /b/. See this paper for more details: "Word recognition and phonology: The case of
English coronal place assimilation," by David W. Gow Jr. and Bob McMurray).
Example: "bad god" might be pronounced like [bæggɒd].
The sound /t/, which is the voiceless equivalent to /d/, may also assimilate in place. It may turn to /p/ before /b/ or /p/, or to /k/ before /g/ or /k/.
The other stops /p, b, k, g/ do not generally assimilate in place across word boundaries.
Almost any consonant in English may be devoiced due to assimilation. For stops, the way this works is that a voiced stop like /p b g/ is pronounced with less voicing (but without aspiration) when it comes before or after any voiceless consonant (including /t d g/, but also including fricatives such as /f s ʃ/). This devoicing also often occurs at the end of a phrase.
Even though a devoiced /d/ sounds basically like [t], the convention is to write is using a devoicing circle diacritic as [d̥] because it is usually distinguished from the English phoneme /t/ in other ways. For example, vowels are longer in syllables that end in voiced consonants than they are in syllables that end in voiceless consonants: we can write this in transcriptions with the partial length diacritic /ˑ/. Also, English voiceless consonant phonemes like /t/ are actually pronounced as aspirated consonants like [tʰ] at the start of stressed syllables (and possibly some unstressed syllables); devoiced voiced consonant phonemes are never aspirated like this.
- "hat god" might be pronounced [hækg̊ɒˑd̥] or [hækg̊ɑˑd̥]. There is regressive place assimilation of /t/, making it be pronounced more like the following velar consonant, so we can transcribe it as /k/ here. There is progressive devoicing of /g/, making it be pronounced like an unaspirated [k] sound, which is still distinct from the aspirated [kʰ] that would be used for the phoneme /k/ in this position.
- "mad king" might be pronounced [mæˑg̊kʰɪŋ]. There is regressive place assimilation of /d/, making it be pronounced more like the following velar consonant. There is also regressive devoicing of /d/, so we use the devoicing circle diacritic. Overall, the /d/ here is likely to be pronounced like an unvoiced [k] sound. But the length on the preceding vowel will clearly distinguish it from the phoneme /k/, which would make the preceding vowel short.
- "big dog" would be pronounced as [bɪˑgdɒˑg̊] or [bɪˑgdɔˑg̊] or [bɪˑgdɑˑg̊] . We do not transcribe place assimilation because /g/ is not as prone to assimilation as /d/ is. We do not transcribe devoicing because both consonants are voiced.
So to return to your examples:
Dad kicks my ass.
The final "d" in "Dad" cannot be dropped here. Regressive place assimilation is possible, however. With assimilation, the first two words "Dad kicks" would be transcribed [dæˑg̊kʰɪks].
Dad loves my mom.
The sounds "d" and "l" are similar. They already have the same general place of articulation and are both voiced. The "d" cannot be dropped. I am not sure if some accents might assimilate the /d/ to /l/, but in any case, this is not a mandatory assimilation. I would recommend just pronouncing the first two words as [dæˑdlʌˑvz].