thyratrons or vacuum tubes for relay control
To 1st and 2nd problem you need a simple flip-flop. You can make it from two triodes or one dual triode. Look at this schematic from DRTE Computer site:



Outputs are on anodes of tubes. One output have high level and other have a low level, trigger pulse swap levels of outputs. There are many vacuum-tube flip-flops like this one, similar in construction, and all of them have its origin in first ever know digital circuit: Eccles-Jordan flip-flop.

Solution to the 3rd problem is single-shot multivibrator:



Output is on the A2 anode. Values of R and C connected to A1 anode gives You a time of output pulse duration. This circuit comes from P.Neeteson book "Vacuum valves in pulse technique", page 117. Another circuit You may found here: http://mysite.du.edu/~etuttle/electron/elect36.htm

There are also other solution to control relays without transistors and also without tubes - You may use only relays. But I suppose that it will be hard to find old "non-electronic" (fully electro-mechanical) pulse-trigerred and time delay relays - eBay and flea-markets should be Your friends.