Dear Larry,

very strange. The CAPTCHA was introduced between 1.6.3 (2006-06-18) and 1.7.1 (2006-12-06) - I don't have all versions in my archive… According to the timestamp I suspect it 'came out' in v.1.7 around 2006-06-25. But since you are actually seeing the checkbox in the Admin section - it must be 'there'…
Do you have access to WebMySql at your webspace?
If yes, back-up the database (from the Admin menu, copy the entire stuff, paste it to a text-edior and save it somewhere on your local machine), connect via WebMySql to your databases, open the forum, open the table 'forum_settings' > Browse
If you are using the current (1.7.5) version, there should be 76 rows in the table.
Settings should be:

row #       name         value
 69    captcha_posting     0
 74    captcha_contact     1
 75    captcha_register    0 
 76    captcha_type        0

According to your list, you are also missing an entry 'stylesheet_file'.
Please come up with your results.
Are you really sure, you are running a clean 1.7 version - and never have replaced just part of the scripts, etc.?