I'm a US native speaker. I understand all of the phrases (I think!) except for "thermal bath."
"Local school" would usually be used in the plural, to refer to a school system. "We have good local schools." The school four blocks from our house, that our children used to walk to, is "the neighborhood school."
An unpaved, irregular, winding, path, going through woods and across fields, is a "trail." An engineered, flat path in a city, that does not fit the word "sidewalk," is a "walkway" or "path."
A child pulls a "wagon" behind her. "The word "toy" would not be used; it would be clear from context. The phrase "little red wagon" is a colocation. Thus, "we repainted the rusty old wagon for our grandchildren," versus "the Mormon pioneers crossed the desert in covered wagons."
It would be natural to say "I've packed a lunch for you" or "Your lunch is packed" or "Pack your own lunch." The phrase "packed lunch" is perfectly clear, I'm not sure it is a stock phrase.
I am not sure I understand "thermal bath." A "Jacuzzi" (trade name used as a generic name), or a "whirlpool bath" is a big tub that massages limbs and muscles, with vigorous stream of strongly pumped water. "Hot tubs" or "spas" are big enough for several people to sit in, relax, and soak. (There are always discussions beforehand about what, if anything, should be worn; the only times I've been invited to use one, the dress code was "bathing suits.")
""Measuring tape" and "tape measure" are equally common. "Measuring tape" makes me think of a loose tape, without any case or mechanism. "Tape measure" makes me think of a little case, from which the tape can be pulled, with a spring mechanism to retract and rewind it back into the case. Measuring tapes can be short tapes of cloth, to measure people for clothes; or long, thin, steel tapes, used by builders and carpenters to measure rooms and furniture.