I can't disclose the full circuit because of an NDA but here is the antenna input section for a similar receiver. This one includes a capacitor in series with the antenna connection, which presumably yours doesn't if you're able to measure a DC resistance:
So when you're measuring the DC resistance the inductor in the parallel LC circuit will appear as a short to ground, but of course that won't be the case at the RF frequencies of interest. So don't try to remove it because it'll compromise receiver performance and you shouldn't need to take it into account when calculating the antenna length.
The PCB and pin lengths may have a bit of an effect of the optimal antenna length but if you're keen enough probably the easiest way to tune it without any test gear is to start with the antenna a bit longer than theoretically required and trim it down until you get the maximum range. That's assuming you're using wire for the antenna that can easily be replaced if you end up making it too short before finding the best length.
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
Best Answer
It appears the antenna is on top of the board as shown. It is the land on the outter left hand curved edge. It is also mirrored and connected with vias to the bottom. It is connected by a capacitor and has two holes drilled in it and further down in the middle, has 3 holes. The photo darkens at the bottom so it is hard to see if it ends in space or connects to something. On the top of the board, cut the land with maybe an exacto knife just before the two holes by the capacitor and solder an equivalent length wire (or 2X, 3X etc) by the capacitor. That's my best guess from what I see. If it doesn't do the job you can scrape the surface clean on the land and resolder it bridging the gap.