Das Problem liegt nicht beim Timer 0, sondern Timer 2, der bei der IRremote-Bibliothek verwendet wird. Timer 2 gibt es bei den Tinies nicht, also muss die Library umgeschrieben werden.

Auf der von Dir verlinkten Seite ist das GitHub-Repository verlinkt, das den veränderten Code enthält. Wenn Du den in Dein libraries-Verzeichnis packst, solltest Du Deinen Code kompilieren können, allerdings heisst die Bibliothek nicht mehr IRremote.h, sondern irrecv.h, das Interface ist nicht mehr objektorientiert, sondern plain C. All das ist notwendig, um Platz zu sparen, damit der knappe Speicher des Tiny 4313 (4k) genügt. Der ATtiny85 hat zwar 8k Flash, aber willst Du deswegen die Library nochmals umschreiben? Freu Dich, dass Du etwas mehr eigenen Code beisteuern kannst.

Das Programm verändert die Helligkeit einer LED beim drücken einer Taste auf der Fernbedienung.

... plus gibt es den Wert auf der seriellen Schnittstelle aus. Nimm das noch heraus, sonst kriegst Du mangels USART auf dem Tiny Probleme.

Da kann ich nur Vermutungen aufgrund der Bilder anstellen. Ich würde folgende (zusätzliche) Verbindungen machen:

Arduino	Display
11	MOSI
12	MISO
13	SCLK
4	CS (the one below MOSI)

Wenn Du das so verdrahtest, kannst Du alle Beispiele für den SD-Karten-Zugriff verwenden (beim Ethernet Shield ist der CS für die SD-Karte auch auf Pin 4).

Das Display wird mit SPI angesteuert. Im Beispiel wird dafür nicht der Hardware-SPI-Port genommen, sondern einfach beliebige GPIO-Pins (Pin 4-8). Du kannst nach dem Beispiel einfach Pin 4 bis 8 nehmen und das Beispiel-Programm unverändert übernehmen oder Du schliesst das Display an den Hardware-SPI an (was deutlich schneller ist) und nimmst das Beispiel "graphicstest_hispeed". In diesem Fall (graphicstest_hispeed) müsstest Du folgende Verbindungen machen:

Arduino	Display
10	CS
9	DC
8	RES
11	SDA
13	SCL
5V	VCC
GND	GND

Die ersten 3 kannst Du auch auf beliebige andere Pins legen, Du musst dann einfach die #define am Sketch-Anfang ändern. Die letzten 4 sind fix.

Make the Arduino look like a bluetooth keyboard to enter the data in text.

This may be difficult because most bluetooth modules available for the Arduino don't support the HID profile but only the serial profile. If you take the more expensive modules you may end with a price higher than the Kestrel station.

I'd try to contact the developer of Shooter (seankndy) and ask him what the plugin interface is. I'd bet that this is a much more reliable way to connect to his app.

My reference document is from here: https://mysick.com/saqqara/im0007888.pdf, where is your's?

There is only an output voltage (nothing about current output), so if in doubt I'd use the voltage divider.

The product description I found on the internet says analog output on Q_A is 0.15...6V, so a simple voltage divider (2 resistors) to an analog input should do the job.

I guess the Arduino part is possible and feasible but I doubt that you can access the variables of another program on an Android device (would be a big security hole).

If you tell us what functionality you wanna use of that big commercial app, we may have a solution how to do this stuff without using that app and probably even completely within an Arduino application.

Kann ich das Display auch ohne dieses Module benutzen ?

Ja, das Sensor Shield hat keine aktiven Komponenten, nur die verschiedenen Pins anwenderfreundlich auf Stecker raus geführt.

Wenn Ja,wie schließe ich es an?

Wenn Du die Beschreibung postest, wie es an das Sensor Shield angeschlossen wird, sage ich Dir, wie Du es ohne anschliessen kannst. Ich kenne das Display nicht und nur vom Foto ist das schwierig zu beurteilen.

I didn't expect it to be readable, nevertheless can you post the results?

Interesting would be the results of the hardware serial 8N1 as well as the results of the SoftwareSerial (8N1 too) with inversion activated (that way we have both common types of start/stop bit signaling).

May be a silly question, but do you have a scope to attach to the line to see the signal pattern?

I am really curious as to if anyone has done something like this before and wouldn't mind giving a complete novice some advice?

Cannot help you with that, not my area.

Can I correlate two serial inputs into a single file through the Arduino?

You can but to keep you from problems with software emulated serial interfaces I'd suggest you get an Arduino Mega 2560 which has 4 hardware serial ports.

I have three example programs, one for each device going into the Arduino (transducer, SD card reader, GPS). Do I need to upload these separately? Or should the program be one continuous line of code?

An Arduino can only run one program at a time. So if you wanna have the functionality of the three programs you have to integrate them into one sketch and upload that.

So can i use the arduino for configuring the bt module?

Yes, you can, but you have to use a level converter (can be a very simple one) because the module is running on 3V3 and not 5V as the Arduino.

I mean that connect arduino on usb to the pc while it is on(it uses tx,rx pins 0,1 for communication with the pc), and also connect the bt module to the tx,rx pins 0,1; or the arduino's spi pins. Does it work, and can i configure the module that way?

No, because when you connect the Arduino over USB the USART on pin 0 and 1 is used for that communication.

Or connect the pc on mega's serial port 1, and then redirect the port1's messages to the port 2,  on which the bt module is.

This is possible and the recommended solution given your parameters. Easier than using Serial1 is using just Serial (the USB connection) for the PC link and connecting Serial1 (pin 18 and 19) to your BT module. Take care that the Arduino's TX pin will have 5V which could fry the module. Using a simple voltage divider may on that pin may be enough though.

I guess the problem is the network setup.

At home you have let's say an IP of 192.168.1.44. The Ethernet.begin() method then extracts the non-specified information about DNS, gateway and network mask from it, this way:

DNS and gateway is same IP with last byte set to 1: 192.168.1.1
Network mask is always 255.255.255.0

So let's say the school's fixed address is 10.2.4.67, then the Ethernet class will assume the following values:

Gateway/DNS: 10.2.4.1
Network mask: 255.255.255.0

If these values are correct for the school network, the Arduino should work. But I guess that they are not correct and that's why your Arduino fails probably. Try getting these values (from your notebook or from the school's network admin) and specify them explicitly on the Ethernet.begin() method.

Have you tried to use a non-inverted 8N1 serial interface to read the values? From the information you provided and that is available I would also assume that it's 7E1 but you can read that with 8N1 and AND every byte you get with 0x7F, that gets equal results for the reading side.
As for the inversion: I guess you're referring to the definition below the table 4-1 in the linked document where  SPACE is defined as >4V and MARK as < 1V. No word is said though about the levels of the start and stop bits. This could mean that you have to read serially without inversion and just XOR the value with 0xFF (NOT operation) to get the bit values inverted but leaving the start and stop bits on their usual level. I hope you understand what I mean.

On a standard Arduino UNO the RX/TX pins ARE the serial monitor. You might have to describe what your problem is and not what you think might be a solution.

Do you have a datasheet for the device you're trying to read? Perhaps we have to adapt other things like activating the pullup on the RX line or inverted start/stop bit logic.