((Hugs), that sounds really tough.
Ignore what people are saying and go with your instincts. He might just be a hungry baby trying to catch up, and you can't spoil him - or know exactly why he wants feeding so much at this stage. It does pass, scarily quickly for me this time around.
Wrt sleep, I had a non-napper who is still really hard to get to sleep at times. But a routine and good sleep habits emerged over time. Just start with the end in mind, if that makes sense? At three months we still had all naps in the sling (exhausting!) as he wouldn't have slept otherwise, but we gradually made the move to the cot.
Don't worry, enjoy him being little as much as you can.