I have build quite a few wireless sensors and remote controls based on those cheap 433Mhz modules and tiny85. Here are my observations:
You don't need to power the wireless module from the microcontroller pin in order to save power, it consumes near nothing when the data pin in low.
When the data pin is high (it transmits) the power consumption is about 14mA, so it consumes about 7mA on average to transmit a message consisting of ones and zeros.
You really need an antennae, just solder a straight 17.2cm wire to it. You can just loop it around the perimeter of the remote if you can't keep it straight.
3V power supply is quite low, when your battery goes down to 2V it's even worst. I use a charge pump voltage doubler in my design to power the transmitter. It's powered down in sleep mode, I turn it on before transmission and turn if off after.
The button cell battery can provide only few mA and if you use the voltage doubler you will also double the current drawn, so you need to use a larger cap on the output to accumulate enough energy for one transmission burst (you can't continuously transmit)
I made an RF library specifically for those transmitters
https://github.com/cano64/ManchesterRF
Check the example codes for transmitter and receiver.
About sleep mode. You can only wake up the tiny85 using hardware interrupt, and only when it goes LOW, tiny85 has only one HW interrupt pin (pin 7, PB2) so you can use only one button directly to wake it up and it must be pulled HIGH when open.
Here is how to use two buttons for wake up using two extra pins.
Wire it like this: [Pin1] -- [btn1] -- [PB2], [Pin2] -- [btn2] -- [PB2]
Before going to sleep: set Pin1 = LOW, Pin2 = LOW, so when you press any of them it will wake up the microcontroller.
After Wake up: To determine which button is actually pressed, put Pin1 HIGH Pin2 LOW, and check the state on PB2, then switch, Pin1 LOW, Pin2 HIGH and check the state on PB2.
It takes only a fraction of a second to wake up and the user will still be holding the button.
PB2 must have internal pullup enabled or use external pullup
There are two reasons why your approach is off: Loading Effects and Series Resistance in the battery itself. Both will throw off your calculations if you don't consider them.
By connecting the circuit to the device you load the battery down (current flows out of it). Previously with the potentiometer not connected to anything, current was flowing already, you can measure this with a multimeter if you don't believe me. By loading down the circuit even more current is required then, so you are getting an even bigger voltage drop on the potentiometer because more current is going through it now. The voltage that's left is what goes to your device.
The effect of the resistance within the battery will cause your voltage to be lower than expected when high currents are required in your circuit. These high currents must leave the battery and cause a voltage drop at the battery terminals. It's not as big of an issue as loading effects but it's something to be aware of.
Here is a more accurate model including these facts. (Ignore the resistor values)
simulate this circuit – Schematic created using CircuitLab
You will also come to learn that a potentiometer is a bad design for dropping to a specific voltage! It's basically the reason you are experiencing loading effects in your circuit.
A better design would use a voltage regulator such as the LM7812.
However in the meantime, and perhaps the shortest answer to this question is: Tune the potentiometer output voltage to 12V while it's connected to the jacket.
Another fun thing that might happen is this voltage may drift as the wires in the jacket heat up and the effective resistance of the jacket changes. This is a discussion for another day.
Best Answer
First off be very careful when dealing with mains power! You can really hurt yourself or start a fire.
You can easily make one with a microcontroller, relay and several other parts (the most complicated part would be getting 3.3 or 5vDc, and that's not too tough.) I designed a similar unit for my work.
To get you started, you should look into using a simple microcontroller (MCU) board like an Arduino to keep the time and switch the relay (turn the outlet on and off.) It would be connected something like this:
The Arduino can keep the time fairly accurately (enough for this type of application, probably off by around 5 to 10 mins a year,) but you could use a real time clock IC if you want to be more accurate. If you're familiar with MCU's you can take a look at Atmel's real time clock app note, it's a bit complicated if you don't know what your doing; and you don't need to go that route if you use the Arduino's software.
You would probably want a display and some buttons to see and set the current time, and on and off times. A 7 segment display (like in the picture below,) would work or you could even use a cheap LCD display like this uses. However you can do without them if you just want to set it all on a computer via USB.
Another thing to consider is a battery backup to keep the time, and other settings saved would be nice too, however you can go without it if you want and you can save the time to the eeprom every 10 mins so that if the power does fail it will come back relatively close to the correct time. You don't want to save the time too often since eeproms have a limited amount of writing before they stop working (in an Atmel AtMega, I think it is ~100,000 times.)
Here are two walkthroughs on what's involved (2 links but the same project I think, I didn't read through to see if they are exactly the same or not though.) Arduino controlled outlet and Controllable Power Outlet.
Here's a picture of a similar device I designed for my work.
(The processor and power regulation parts are under the 7 segment display.)