There are a couple of relays that may be implicated. The electric fuel
pump is switched ON and OFF by the "fuel pump relay". If it gets tired
and fails to stay ON the engine's fuel supply is cut off and it will
stop and may not restart. The second relay causes odd behavior but not
such a refusal to start; it's called the "over voltage protection relay"
and has a fuse on its top. Either could be the cause, the fuel pump the
more likely of the two.
It could also be that the reason this car was for sale was this
intermittent stalling problem. If you suspect that may be so, I suggest
you have a good M-B independent shop look into the stalling for it won't
go away so it must be fixed. Some cars of this vintage had bad wiring
harnesses that shorted out - the ecologically correct insulation dried
out from the engine's heat and crumbled allowing the wires to short. Any
shop can determine if this is your car's problem, unfortunately it's an
expensive repair but then it's done, not to reoccur.
Then there's the mystery alarm. However, I'd discount the alarm because
the engine was running when it stalled; it's not like it wouldn't start
Whatever course you choose to deal with the problem, always ask the cost
BEFORE agreeing to a repair for this may well be a "let's try this and
if that doesn't work we'll try the other approach" situation. Some shops
will install an new electrical component and reinstall the original if
the original is proven to be OK. Ask if the shop will do that so you
don't needlessly buy a lot of parts.