All of the sentences but the 2nd are fine, but you are incorrectly mixing tenses (past and present). You need to decide if what you are talking about is in the present or in the past. Also, 'amongst' is better here than 'among', but native speakers will often mix the two so don't worry to much about this.
“Among all things, the gift I MOST wanted my mom to give me is a new phone" should be either:
"Among all things, the gift I most wanted my mom to give me WAS a new phone" or
"Among all things, the gift I most want my mom to give me IS a new phone".
"Among all things, the gift I MOST wanted to HAVE my mom give me is a new phone” should be:
"“Among all things, the gift I most wanted to have my mom give me WAS a new phone” or
"“Among all things, the gift I most want to have my mom give me IS a new phone”
"Among all things, the gift I wanted to have my mom give me (THE) MOST is a new phone” should be:
"Among all things, the gift I wanted to have my mom give me the most WAS a new phone” OR
"Among all things, the gift I want to have my mom give me the most IS a new phone”