Electronic – Why does the MCU work better when powered to Vcc and PA1=D2 instead of Vcc and GND

All in- and outputs on most digital devices today have clamp diodes to the power rail. These diodes are there for protection of the device, to prevent a pin from having a higher or lower voltage than the supply rail. What you are experiencing here is that you actually power the DAC through its data pins and the respective protection diodes. Although the device seems to work, it is not designed for this mode of operation. For example you may easily exceed maximum pin current or maximum protection diode current. It is often unspecified if and how a device fails when doing so.

^{simulate this circuit – Schematic created using CircuitLab}

And below is what happens when you connect two or more input pins, remove the regular power supply (and add the decoupling cap). If you look carefully, you'll notice that the four diode clamps form a diode bridge rectifier (Graetz circuit). According to the article linked in the @PeteKirkham comment below, the external cap isn't even necessary. Probably the parasitic capacity in the device is in that case already enough to power the device.

^{simulate this circuit}

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