Should I tell the truth to my manager who always says “tell me anything” but is offended by little criticism?

ALWAYS tell the truth.
Understand. Each of these have special purpose and value:
Being honest, truthful
Giving feedback
Offering “criticism”
To Opine/give opinion (imposed, or shared helpfully) Any of these, can be either “welcome and helpful” OR “hurtful and offensive”. WORDS matter, regardless of your intent. It’s a form of tactfulness and diplomacy.
Even the most sincerely “honest”/truthful comment or message can (unintentionally) be presented offensively.
Most Managers seek feedback about work, or methods. Personal criticisms and “complaints” (without solutions) are rarely helpful. Choose your words thoughtfully, and carefully.
IME, these tend to make people (BTW, not just managers and bosses!) defensive: Stupid, improper, wrong, dumb, should/shouldn’t, statements that begin with “you”, always, never.
Work to use objective terms: “X could be…more helpful/practical/effective/a better idea!…”, or “Y could be …less time consuming/labor intensive/costly.

Yes, but keep in mind that when and how you say something are just as important as what you’re saying. How much you’re saying is also important…some things need multiple conversations or aren’t all that relevant. Divide those up or get rid of them altogether.
Deliver feedback that is honest, authentic, and timely (which means sooner than later in this case). Also, have a focused, emotionally intelligent approach that doesn’t come off as infantilizing. Managers usually will respect people they see that can think for themselves, even if they don’t 100% agree.
There’s a saying that’s something like “being agreeable is learning how to tell someone to go to hell and have them thank you for it.” That’s applicable here to some extent, you need to learn what your manager responds to and then serve it up in that way…they might thank you for it.

Yes. One way to start out is by mentioning a few general issues and then throwing in one that may be a bit controversial. If they “react” to the latter, I’d say, “I’m sorry. I thought you wanted to hear it all. I won’t mention this matter again.”
I’m guessing, if they’re a good (if sensitive) manager, they’ll say something like: “No, go on. I need to hear that stuff too.”

Good luck.

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