Is it common among native speakers to barely pronounce the first letter/phoneme of a word? The voice here barely pronounces the 't', and I'm finding I have to listen very closely to hear many of the prepositions as they tend to be half pronounced and run into the next word.
It is common to connect prepositions with the subsequent word. In all languages I know.
You should not skip any letters except some specific cases (jsem -> sem, jsme -> sme). You may find it difficult to hear some consonants, but they have to be there and a native speaker would immediately find it missing or would not understand you.