Not a lighting expert but:
(i) It won't be a rheostat controlled as such - this term means that a variable resistor is adjusted so that the amount of power it "takes" from the circuit is variable. So:
(a) at high resistance not much current is flowing in the circuit and the bulbs are dim. Bulbs and rheostat are dissapating little power and are cold.
(b) at medium resistance a decent amount of current is flowing and the bulbs are on, but huge amounts of power are being dissapted in the rheostat and it will be very hot. Hence they are inefficient and went out (in most cases) with the ark.
(c) at low resistance a lot of current will flow, the bulbs will look fully on and the rheostat will be cool.
(2) The confusion in terms come from the fact that in most dimmers a variable resistor (rheostat) is used to control some form of electronic switch (normally a triac) so people assume it is the same as the early versions (pre - 60s?). It ain't.
(3) The newer version generally employ a form of phase control (ie not all of the cycle (sine wave)) is available to the load. Many modern lights that need a voltage to strike don't get on with this. Likewise with some transformer (low voltage) based systems.
(4) Can't think of a domestic situation where you would connect any lights in series - one out, all out.
(5) Suspect you have a wiring fault or something based on (3) at the root of your problems, but there my generic knowledge runs out.
...and I bet someone has given the correct answer in the time it has taken to write this!
