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Dries,

Here's what I hope is a helpful answer to your questions about the Bitx20

Yes, you could use FT-37-43 cores on the transformers.
Just duplicate the transformers used on the Bitx40, 10 turns trifilar:
? ??

Carrier suppression on transmit mostly depends on T3 and the two nearby diodes
being wired correctly, also the nearby 100 ohm pot must be adjusted for best suppression.
The crystal filter will then improve on this somewhat, assuming the BFO is adjusted correctly.

Curt's suggestion to check the BFO placement is a good one.
Have you tried adjusting the 22pF trimmer in parallel with L3?
There's a whole paragraph about tuning in the BFO on? ?
? ??
starting with the words "On amateur bands above ..."

Here's a technique for debugging a receiver that might prove useful:
??
Farhan uses a short piece of wire as an antenna, perhaps 1 meter long.
It picks up houshold QRM, and he injects that into various parts of the radio,
listening to the result out of the speaker.? Perhaps this will help.

Once the receiver mostly works,?you might verify the crystal filter shape.
To do this, measure the audio coming out to the speaker using a scope
while tuning through a strong carrier in one of the bands.??
Ideally, you would start hearing it around 300 hz on one side of the carrier,?
it shouldould quickly rise up to full volume and stay there till around 2500 hz,
at which point the volume should rapidly fall into oblivion again.

I am not quite sure what to say about all of the "mixing frequency products"
that you hear in the receiver.? I don't know how clean this radio was,
but they should not get in the way of casually tuning in stations across the band.
It could be just a matter of a wiring error or bad part.
Or it could depend on how the parts were placed on the board.

When building the crystal filter, it is best to choose a set of?
four 10mhz crystals that match in frequency within perhaps 100hz
when plugged into an oscillator, using a frequency counter.? ?
It may be necessary to do this if the above test of the filter shape looks wrong.

We can worry about Q14 and the 2n2218 JBOT power amp
after you get the receiver working properly.
?
If you get stuck, you can certainly ask questions here.
But best would be to find a local club of radio amateurs who know
about working on radios and have appropriate test equipment.
Perhaps someone there would like to help.

Best Regards,
Jerry Gaffke, KE7ER