Minimum score for 7 letters and what beats it
The minimum score for a seven letter word if you can place the word anywhere is 55.

e.g. 


  RETiNAs C2 across. (lower case indicate blanks)


The score is 5 for RETNA, 0 for the blanks, and 50 bonus for using all 7 letters. The word would lie on a TL square, but since this is covered by a blank, it gives no bonus since 3x0 = 0.

This could be beaten by a three-letter word e.g.


  ZAX A1 across


The score is 19 x 3 = 57. I can't think of any common three letter word that would fit here, but ZAX is valid in both TWL and SOWPODS dictionaries. For common words, I suspect you'd have to go to four letters and something like QUIZ.

The largest possible score for a two-letter word, before bonus squares, is 11 (QI, ZA or ZO), so you couldn't get higher than 33.

While these match the criteria in your question, they could never occur in isolation in a real game since they would either have to be opening moves across the central DW square, or joined onto existing words.