"لا أَعْرِف اِسْمهُ."
Translation:I do not know his name.
Because اسم does not start with hamza (أ or إ) but with wasla ٱ.
As a result, there is no guttural stop at the beginning of اسم but you contract it with the previous word. If the previous word ends with a (short or long) vowel, the initial ا even becomes silent, e.g., باسم الله bi-smi-llah in Allah's name (actually contains two silent alifs).
There are other words than اسم which start with wasla and not with hamza. The most prominent example is the definite article الـ. So, yes, it is particular to the word but not only to اسم.
A hamza should be always written (otherwise, it is a spelling error, as far as I understand). A wasla is usually omitted so that you see a bare alif at the beginning of a word.
As to the pronunciation of this sentence, yes, you hear lA 2a3rif-ismuh in dialects. In MSA, however, it would be lA 2a3rif-u-smuh.