Don't believe everything you see on YouTube or online. I have seen some utter rubbish online and lots of people reckon it's really good information.
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Cordon Blue finches require live food if you want to breed them. So do virtually all African finches. Australian finches on the other hand don't need live food and do really well on green grass seeds when it comes to breeding. This should make your mum happy (no bugs in the house).
The weavers should have an aviary at least 3 metres (10 feet) long and should not be kept with Gouldians or other small peaceful finches because the weavers can get a bit aggressive during breeding. Weavers actually do best in huge aviaries with multiple pairs of birds so they can breed in colonies.
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Most Australian finches do best in colonies and breeding results can be reduced if they are kept with other species. Gouldians and owl finches (double bar finches) should be ok but the Gouldians are significantly bigger than the owl finches and might bully them a bit.
The Bengalese (society) finches are big enough to live with Gouldians but might decide to take over the other bird's nests and rear their young.
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You can grow bamboo, Australian bottlebrush, Eucalyptus and Acacias (wattles) in pots and put them in the cage for the birds. They are all safe for birds and quite easy to grow. They have tough leaves and do well indoors or outdoors (in most climates) and would provide you with some natural plants in the cage.
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Have you thought about making your own cage/s?
You can buy slotted angle iron from any hardware store and build a frame with that. The frame gets held together with nuts & bolts. If you make the frame 6-10 foot long x 2 foot wide x 6 foot high, you could divide it into 3 sections (going up). This would give you 3 separate cages that were 6-10 foot long x 2 ft wide x 2 ft high. You wrap the whole thing in wire and put some shelves on each level and away you go.
This size cage would let the birds have flying room and you could have different birds in each cage, but it would only take up a small amount of space.