dditch,

Yes, the hoses are correctly installed, and the engine is quiting due to lack of fuel, it is highly possible you have the engine internal fuel passages clogged, you will have to disassemble the engine completely and flush it, use alcohol, for extreme gum I use paint thinner (the paint thinner will dissolve the gum and all parts made of silicon like the regulator diaphragm so be carefull), and then try to blow all passages with compressed air.

After all that is done, you need to change all the seals, you show only the diaphragm side of the regulator, but the poppet valve (plunger) might be stuck or worn on the adjuster screw side, you need to take this apart also to clean it, get a new diaphragm, it looks in your picture like it is dented or marked on one edge, and if it leaked fuel I assume is because of this, the plunger and diaphragm are cheap so get new ones it might save you a lot of time, also, you need to change the O rings on the rocker arm push rods tubes.

The regulator should be flush with the housing after you reassemble the engine, do you have the engine manual? if not I'll email it to you.

Regards