な-adjectives

A sentence from a story about Doraemon:
未来のいろいろな便利な「ひみつ道具」をポケットの中に持っています。

I have a quastion about connecting of the na-adjectives いろいろな便利な. I've learned that the first na adjective should end with で, then follows the next adjective. Like いろいろで便利な. Or are there rules when two adjectives can be connected with な? Can i-adjectives also be connected without -kute at the end? おいしいいろいろなケーキ? おいしくていろいろなケーキ? Thanks

いろいろ is a bit special. Unlike 便利な or おいしい, it doesn't describe the characteristics of each object (tool or cake). It describes the characteristics of the whole unit. Think about "many kinds of" or "various" in English. These words are for the whole "useful tools" as a unit, not for each tool. You can think that it actually means いろいろな種類の便利な道具.
cf.
いろいろな便利な道具
便利ないろいろな道具

安価で便利な道具
便利で安価な道具

いろいろなおいしいケーキ
おいしいいろいろなケーキ

おいしくてかわいいケーキ
かわいくておいしいケーキ

(Note that the order 便利ないろいろな or おいしいいろいろな are acceptable and actually used, but it could sound a bit awkward.)

Thanks, I see what you mean. So, each time いろいろ is used, other adjectives describing the same unit as いろいろ don't change their form to -kute and -de respectively regardless of their position (before or after いろきろ), is that correct?

Yes, that's right. Incidentally, ほとんどの has a similar function. This adjectival phrase also describes the characteristics of the whole unit. Furthermore, both いろいろ and ほとんど can be used as an adverb.
e.g.
いろいろな便利な道具を持っている。
便利な道具をいろいろ持っている。

ほとんどのおいしいケーキは売り切れた。
おいしいケーキはほとんど売り切れた。

(Note that おいしくてほとんどのケーキは売り切れた。 is valid, but おいしくて is interpreted as a cause/reason clause to explain why most cakes were sold out as the same meaning as おいしいので, not modifying ケーキ.)