Measure the internal width of that old road rim - it's probably 15mm ??
That Alex rim seems to have a 16mm internal rim width, if I'm reading the specs correctly.
Internet recommendations vary, but most say a 17 mm rim width is needed to mount a 38 mm tire.
(It probably depends on the tire as well? Casing flexibility?)
About the bulge at the valve - are you sure the tube was pushed 'up' inside the tire? I've learned to make sure the tube is freely moving at the valve by pushing the stem 'up' into the tire before inflating. This seems to prevent getting the tube stuck under the tire bead.