The DHT11 Sensor that you are using is a poor quality sensor that only has 1D resolution, so the best you can get
is readings like 22.0 D or 30.0 D .
You cant get fractional parts of the temperature.
You need a better sensor to get fractional readings, like a DHT22.
The DHT22 and the DHT11 are similar in pinout only, the decoding of the data is differant.

What you are trying to do is not that easy, and quite dangerous if you get it wrong.
You have to synchronise the pwm drive to the solid state switch , with the zero crossing points in the AC waveform
so that the switch gets turned on for equal amounts of time in both the positive and negative directions.
Most applications like this that are used in heating ,use what is called zero cycle switching which simply turns on or off complete AC cycles
into the transformer.
This way there is no chance of a DC component through the transformers primary.
If you are not familiar with working with mains voltages , I would not be doing this sort of project.

Need to work on your maths.
100 amps and 2.6 mohms = 100 * 100 * 0.0026 = 26 watts.
Fet will need a heatsink.

For people who may not understand how these cheap 433 radios work , hers a very basic explanation.
The radios use amplitude modulated keying , which simply means a logic 1 turns the transmitter on , and a 0 turns it off.
All the Transmitters and Receivers are on the same frequency, normally 433.92 Mhz.
The Receivers have a bandwidth of around 4 Mhz , which means they will receive anything that falls between 431.92 to 435.92.
The receivers will receive anything in this range , including transmissions from other peoples transmitters and in the case of when nothing is being transmitted the receiver just produces digital garbage , which is essentially the digital equivalent of thermal noise.
The receivers know nothing about data rates , baud rates , whether data is serial or not , or anything else, but simply respond to amplitude transitions at their input.
To use them properly, you need to provide some kind of flag sequence at the start.
A flag sequence is a series of 1 0 1 0 1 0 transitions that needs to last a couple of milliseconds.
The purpose of the flag sequence is to allow the receivers AGC (automatic gain control) to adjust itself to the input level of the incoming signal
and to allow the receivers data slicer (the part of the receiver that decides whether the data is a 1 or 0 ) to adjust itself to the 1/2 way point of the amplitude variation.
The data slicer needs transitions to work, ie needs data that continuously changing, so sending a constant series of 1 s or 0 s in a row , will result in the receiver not decoding properly.
Most applications that use these radios use some form of data coding like Manchester or similar coding schemes  to ensure that there are always data transitions , even when continuous data sequences of 1s and 0s are being sent.
The receivers AGC has a fast attack and slow decay characteristic which means that if the receiver receives nothing for more than approx 10 ms  , the AGC winds up and the receiver starts producing garbage again.
There is a maximum rate at which you can send data with these radios , usually around 4800 transitions per second , some will do 9600, but not reliably.
There is also a minimum rate of around 100 transitions per second , as less than this and the receiver thinks there is no signal at all in the off periods and winds its AGC up.
Trying to reverse engineer appliances which use these radios is a time consuming tedious task, and ideally you need a digital CRO that can store long periods of data which you can then analyze.

A common method of sending data is to send a flag sequence , ie lots of 1 0 1 0 transitions and then a sync , which is a short sequence differant from the flags but easily identified, then whatever data you want to send.
This can help when trying to reverse engineer data streans , as the flags are always at the start and dont change , the sync doesnt change , but the data does, so it helps you to know where in the sequence the data is likley to be.

LIPO batteries must not be overcharged, which is around 4.2 V per cell.
Theres nothing in your diagram to stop the battery being overcharged by the Solar Panels.
Otherwise it will work.

The pullup resistor is needed because the data line to the sensor is bidirectional.
The sensor only sends data after its been told to by the Micro .
However, depending on the code that actually reads the Sensor, some Micros have weak pullups in them
that means that the resistor may not be needed.
You would have to study the code thats actually doing the sensor reading to see if this is the case.
The resistor value isnt critical, 10K is a good value.
Also, some suppliers of the sensor send it on a small breakout board, which already has the resistor installed.
If the Sensor is jumping back and forth between degree readings 3 D apart, then there is something wrong with either the code thats reading it
or the wiring , or the sensor itself.
The Sensors are pretty rugged so its not likley to be the Sensor.
They either work properly or not at all.

Whats the accuracy of the store bought sensor you are comparing the DHT11 with.
The DHT11 is a slow sensor, which means that it takes a while to respond to temperature changes.

Its possible, but without more information its not possible to know whether its a sensible thing to do or not.
if the Micro is inside, why not just power it from the mains via a plugpack.
Will be much simpler and easier than any solar solution.

Do the sensors require bidirectional transmission ?
Its relatively easy to make small lightwight unidirectional transmitters, but as soon as you go to bidirectional transmission
the power consumption drastically goes up, as receivers have to be on all the time .
 

Its not only the flywheel that requires mass.
Flywheels dont produce any energy at all unless they are mechanically coupled to an electrical generator, which also adds mass.
Adding mass to a helicopter is the last thing you want to do.

Whats wrong with one of these?
http://www.hobbyking.com/hobbyking/store/__11954__Turnigy_nano_tech_4000mah_3S_45_90C_Lipo_Pack.html
90C discharge rate , thats 360 amps , and weighs 360 G .
The flywheel in the demo weighed 1 KG and was only able to light up some small leds.

You will be far better off using a small battery.
The battery is not for running the Micro over the night, but is there to ensure that in low light in the morning when the Solar Panels voltage rises
sufficiently high to start the ATtiny, there is enough current available to run it.
Solar cells dont have a linear voltage / current relationship.
The voltage with no load rises at a logarithmic rate whilst the current capacity rises at a linear rate for a given illumination.
What happens with no battery, is that the solar voltage will rise quickly to above 5 V , which will cause the 7805 to start operating, but the solar cell has no capacity to deliver any current , so the voltage immediately collapses, and the Micro stops.

Ive made a number of remote operated devices which run off Micros In my case using PICS not Arduinos , but the principle is the same.
You need to know the daily power draw of the device to be powered.
You then need to know the worst case sun hour value for the middle of winter, usually around 2 - 3.
You then have to allow for multiple days of cloudy weather , usually 5 days is a good compromise , but varies somewhat depending on your latitude.
You can then size the battery and the solar panel to suit.
Try and reduce the power draw of the device to an absolute minimum when its not doing anything useful.

You need to know a lot more about the Transmitter.
How does it send data, what type of modulation is used.
Without an oscilloscope, this will be very difficult to do.

The other factor thats missing from the demo is how much power was put in to spinning up the flywheel
compared to what was stored.