Why won't they survive after germination? They came up, very spindly, then they fall over and die. What's the secret to keep them growing?
I don't know specifically about salpiglossis, but what you are describing I would attribute to low light and higher temperatures. The higher the temperature at which something germinates and grows, the higher the light requirements. If no one else chimes in with specifics about salpiglossis, I would try lowering the germ temperature and keep them cooler while growing. This is assuming that you are growing them with artificial light inside.
Falling over and dying could also be damping off, which is a fungal problem caused by keeping seedlings too wet. Falling over on its own can happen when things get spindly because they don't have enough light, but since they're dying too it makes me suspect there could be something else going on too.