Drawing that much current effectively mutes the 47uF capacitor.
It sounds like you want to drive a high current load, motor solenoid etc. 
Driving such loads will likely cause problems with the controller as dips in Vin will effect the Arduino.
If you need external high current capability use 2 or more power supplies with common grounds.

I'd like to be able to detect whether there is voltage across the terminals leading to the thermostat on the hot water tank.

I want to detect that current

Do you want to detect the Voltage or Current?

You can detect the voltage with an opto isolator which is monitored by the Arduino.

I have always made it a policy to always use a separate power supply for the controller and the motor!

If I use 7v as the AC/DC input power (see diagram), then the motor starts to work erratically.

I would first confirm the 7 volt power supply is not being pulled down too low.

SN754410NE H Bridge

This is rated at 1.2 amps 2 amps peak. How much current is drawn by the motor if you connect it directly to 7volts?

Is this all the code you have?

[wiring]:
A4 connects to   RTC: SCL
A5 connects to   RTC: SDA
GND to GND
5V to 5V
SQW does not connect to anything (yet?)

Arduino Uno
A4 = SDA  ---->  RTC   SDA
A5 = SCL  ---->  RTC   SCL
GND ---------->  RTC   GND
5V    ---------->  RTC   5V

For example if you use        T1360280782
You will see                        23:46:23 7 2 2013      on your serial monitor.

https://www.youtube.com/watch?v=wfHCKhXEM6Y

This link has the ZIP file containing the: sketch: Readme: PNG & parts list he is using.
He is using TTL serial communications.

Your display has sample code in the 4th image at: http://www.amazon.com/gp/product/B00A8H6BQ2/ref=oh_details_o02_s00_i00

I believe your Data Sheet for the display is here and it is SPI communications.
http://www.ams.com/eng/Products/Lighting-Management/LED-Driver-ICs/AS1108

Look for BOX or PROJECT BOX
https://solarbotics.com/search/box/
https://www.adafruit.com/index.php?main_page=adasearch&q=box
http://www.jameco.com/webapp/wcs/stores/servlet/StoreCatalogDrillDownView?langId=-1&storeId=10001&catalogId=10001&freeText=project%20box&search_type=jamecoall

4 digit-7 segment LED display

Give us a link to this item.

#include <LiquidCrystal.h>

Why are you using this?

If you haven't already seen the Time library and the included TimeAlarms libraries you may benefit by looking at them:
http://playground.arduino.cc/Code/time
http://answers.oreilly.com/topic/2704-how-to-create-an-arduino-alarm-that-calls-a-function/

Give us a link to your motor shield...

What is the resistance of the Solenoids?
What is the voltage rating of the Solenoid?
If you swap the transistors does the problem swap?
When the Arduino o/p is at 5 volts, what is the voltage at the collector (to GND) of the 2 transistors?
It is hard to see the breadboard wiring, please draw a schematic of the project.

The motor may be a lower current rating therefore it works.
Note: the gain of the 2N2222 is 75 but it can vary.

See ideas for layout:

As a side issue, when it comes to important stuff like flooding etc. you should have a backup circuit.
Something like a float/limit switch independant of your controller.
It would provide a cutoff to the filling of the tank at an extreme level.

Please attach a drawing or schematic of your project.

http://arduino.cc/en/Reference/Array
http://www.thebox.myzen.co.uk/Tutorial/Arrays.html

Code:

//Lets say you have an array:
char dtmf [20]
int i = 0;

setup()
//some code

loop()
{
// some code
          irqState = digitalRead(STD);                //Get the first trigger tone
          if (irqState == 1)
          {    
                dtmf[++i]= mapKey();
                delay(100);
          }

// lets say an # starts iteration code thought dtmf[0] to dtmf[maximum] to process through the values
// don't forget to set i = 0 when you are ready to start capturing again.


} // END of loop

I probably breathe a lot more solder fumes than I should.

I have a small fan blowing them away.