If you pull apart a commercial power line ethernet adapter, you will see that the limitation on size
is the isolation transformer, thats used to isolate the mains side from the low level logic side.
Not much you can do to make this component physically smaller.

If the Rain Guage is the tipping bucket type, its very important that the rain guage is mounted so thats its completely level, and it cant shake around.
The actual tipping bucket is very light, and doesnt take very much vibration to make it tip by itself.
If its not level, then it wont measure properly.
I noticed that in the Sparkfun example, the rail guage is mounted on a pole which is very bad.
They should be mounted close to the ground where they dont get affected much by the wind.

With battery powered projects, the main issue is how long do you want the battery to last, before it has to be replaced.
Coin cells have extremely low capacity, 100 mah or so is typical.
They are mainly designed for backup, watches, anything that draws extremely low current , 10s of uA.

Its simply not possible to make an efficient fridge with peltier coolers.
Their efficiency is far too low , typical Carnot efficiency for a peltier cooler is 5%
compared to around 50% with a compressor fridge.
 

Most small remotes use a SAW resonater to determine the frequency they use, and they arnt adjustable.
If its on 350 Mhz , you will have to source a 350 Mhz receiver or make one.

You need an opamp specifically designed for this purpose like this.
http://www.analog.com/static/imported-files/data_sheets/AD626.pdf

Operates off 5V , but can measure common mode voltages as high as 50 V.
Has fixed gain of either 10 or 100.

The spec sheet is here.
http://www.atmel.com/Images/doc8161.pdf

Sect 28.8 ADC Characteristics.

Vint Internal Voltage Ref   Min 1.0 Typical 1.1 Max 1.2
Seems pretty clear to me.

Its important to read the spec sheet for the Atmega328 when looking at the internal voltage referance specs.
The 1.1 volt referance can vary anywhere between 1.0 to 1.2 volts, which is 20% variation.
Its not a stabilised voltage referance.
If you want a stabilised referance voltage for things like calibrating voltmeters, then you need something like this.
http://www.analog.com/en/special-linear-functions/voltage-references/adr4525/products/product.html
Voltage guaranteed to be stable within 0.02%.
They aint cheap though.

This is a sample of the data stream of an Oregon Scientific Weather station temp Sensor.
Whilst your Weather Station wont be the same , the data format will be similar.
The timebase on your audio card is far to slow to see anything sensible, and what you are seeing is the normal output
of an ASK receiver with no input, ie broadband noise converted into random rate data.
You need a CRO so you can edge trigger the CRO on the rising edge of the first data byte from your transmitter.

Most of the common wireless weather stations use some form of Manchester coding, http://www.atmel.com/images/doc9164.pdf
to transmit the data, as this provides a more reliable form of transmission.
Usually there is a framing sequence of some kind consisting of alternating 1 and 0 to allow the AGC in the receiver to stabilise.
The best way to figure out what is being sent is to get one of the sensors, like a temperature sensor and capture what it sends .
You will have to pull it apart and sample the data on the data pin of the Transmitter Module.
The benefit of looking at the transmitted data is that it gets rid of all the other signals and interferance that confuses what you are seeing.
You are going to need something better than a sound card though to do this, some kind of CRO at a minimum to capture the entire data sequence, as you have to figure out if any kind of CRC is used or not.
The CRC is sent at the end of the transmission.

It seems that you are feeding an AC signal into the Arduino, which wont produce any kind of sensible result.
You need some kind of peak rectifier to convert the signal into DC between 0 and 5V, and also possibly some smoothing.

Nimh batteries dont need balancing as NIMH chemistry can withstand overcharging if the charge rate is C/20 or less.
The battery just gets warm and dissipates the heat.

LIPOs cant be overcharged at all, so the complex charging regime needed.

Any particular reason you want to use Lithium Ions .
Li batteries are good , but as you have discovered introduce a whole lot of issues when it comes to charging them.
Can you use NIMH types instead.
Far fewer charging issues.

Have a read of this , may help to alleviate the confusion.
http://batteryuniversity.com/learn/article/types_of_lithium_ion

Lithium Ion is a just a generic name for a large range of differant batteries which have differant chemistries, with Lithium being the common element in them all.

Appliances such as cameras, lap top computers which use LIPO batteries provide the batteries in a sealed
battery pack , which also contains an intelligent battery charger / balancer, so all the work of looking after the battery is done for you.
If you buy LIPO batteries just as cells connected together, they wont have any battery management hardware built in , so you have to provide it yourself.