seventh: ok, so this one is metaphorical.
Anyway, it’s just the usual, comparing honesty with a precious jewel to take the jewel’s qualities of rarity, preciousness, and beauty and point them out in honesty as well. And then, of course, calling it out of fashion to highlight that, like out of fashion jewelry, it isn’t often used anymore. (according to the speaker, anyway.) The quote also seems to have some nostalgia for the past, as it implies by saying “out of fashion” that honesty was once in fashion, maybe in an earlier generation or era.

Anyway, the first one seems to be equating lying with stealing, meaning the speaker either believes the kind of people who lie will also be the people who steal or sees lying as a kind of thievery.

second: out of context, this seems to be pretty much saying flat out what it means. I’m not seeing anything “deeper” here…

third: looks like it’s just pointing out that after lying, you have to remember what you’ve said, while by telling the truth you need only keep track of what has actually happened. So, lying is more work, whereas telling the truth is natural and requires no extra effort.

“When they speak truth they are not believed.”
Some people are so used to lying that everybody has this image of them always telling lies and when they actually want to tell the truth, people don’t believe them anymore so they have to stick to telling lies. Is like the boy who cried wolf.

“A good memory is needed after one has lied.”
This is true. Because you always have to remember all the lies you told to cover the previous lies. Once you start lying you have to keep on lying and you don’t know what lies you told to cover the previous lies. So have a good memory. Most people end up tripping themselves once they start lying and usually get caught lying.

“A striking difference between a cat and a lie is that a cat has only nine lives.”
A lie only has one life. Because once the truth comes out, the lie is discovered so that’s the end of that lie.

“Dare to be true: nothing can need a lie.”

If you are always honest and truthful, you never need to lie.

“No lie ever grows old.”

Some lies can get out of hand and become huge lies. Lies are always going to be lies no matter what.

“Honesty is a fine jewel, but much out of fashion.”
That is so true. Is hard to find honest people nowadays. So when you actually find an honest person, hang on to them.

“Every man has his fault – honesty.”
Sometimes if you are honest, you finish last or lose good opportunities. Sometimes the people who tell the most lies are the ones who end up getting ahead of honest people. For some people ethics are nonexistent and they are willing to lie, cheat, and steal if that is going to help them get ahead in this world. That is such a shame.