That table looks heavy. Adjective. Correct!
Insistently, he looked at the table. Adverb.
This is grammatically correct, but I'm not so sure about the meaning. If you mean he looked on the table "emphatically, resolutely, firmly", the adverb you want is "intently". Despite the primary meaning of "to insist", "insistently" sounds strange here.
He spoke right. Adverb. Dubious. "Right" as an adverb meaning "correctly" is informal. Common phrase, but some would call it illiterate and insist on "he spoke correctly".
He looks right. Adjective. Correct! But beware: the sentence can also mean "he looks to the right" (direction opposite to "left"; and "right" is an adverb).
The former children appear free regarding their behaviour. The adjective is grammatically correct, but "regarding their behaviour" is not idiomatic. "Appear" is correct, but it is very formal: "seem" may be better.
[And "former" makes sense in two cases: (1) you have already defined two categories of children; (2) you are speaking of adolescents or adults.]
The possibilities are:
The children appeared suddenly in the back yard. Adverb. Correct!
- The former children seem free in their behaviour.
- The former children's behaviour seems free.
Beautiful or beautifully. The choice depends on what is beautiful: the temptation, for which an adverb should modify the adjective, or the lasagna, in which case a comma clarifies the meaning:
You have analysed the grammar correctly, by and large. But as you can see, the choice of words is very important!
- Mrs. Ficara intended to use the homegrown tomatoes to make her beautifully tempting lasagna.
- Mrs. Ficara intended to use the homegrown tomatoes to make her beautiful, tempting lasagna.