CA, small children disturb the davening as they do not yet know how to daven and are incapable of sitting quietly. See Mishna Berura 97:3.
Syag, if the muktza was put there before Shabbat along with something more important or if it was not left intentionally there is no prohibition. Similarly, if the muktza is a devar she’melachto assura, the stroller may be moved for it;s needs or a permitted use (SA OC 310:7-8). How do you define not Shabbat-related? In general, if someone enjoys something that is his oneg Shabbat (Aruch haShulchan OC 307:2-3). I suggest that you review the halachot.