Ill-fortune, hardship, and pileup.
hard is the antonym of easy happy is the antonym of sad
hospital
hot
heartless
Halt.
hate
Haste is a synonym for in a hurry. It begins with the letter h.
Something to do with help: helping, helpfulness
Heavenly is a synonym for tasty.
Another word (synonym) for the word "unpleasant" that starts with an "h" is "horrific". Or at least that is what I could think of.
http://thesaurus.reference.com/browse/boastful There are some good entries here, just look at all the h-words.
The phrase "to lift" means to pick something up. There are several different words that can be substituted for it. Synonyms for it that begins with the letter "H" would be heave, heft, hike, and hoist.