I will answer the two questions separately.
How to make the form of i-adjectives before ございます
Grammatically はよう, ありがとう, めでとう, たのしゅう, おいしゅう in these examples are called ウ音便 (うおんびん) of はやく, ありがたく, めでたく, たのしく, おいしく, respectively. 音便 (おんびん) means the form modified for easy pronunciation.
The actual form of ウ音便 of an i-adjective depends on the vowel before く in the -く form (or equivalently, the vowel before い in the dictionary form).
- If the vowel is -a, ウ音便 is -ou: はやく→はよう, ありがたく→ありがとう, めでたく→めでとう. This -ou is pronounced as a long vowel /oː/.
- If the vowel is -i, ウ音便 is -yuu: たのしく→たのしゅう, おいしく→おいしゅう.
- If the vowel is -u, ウ音便 is -uu: さむく→さむう.
- (If the vowel is -e, ウ音便 should be -you, pronounced as a long vowel /joː/, but see below.)
- If the vowel is -o, ウ音便 is -ou: おそく→おそう. This -ou is also pronounced as a long vowel /oː/.
(I could not think of any i-adjective whose dictionary form ends with -ei, and the only adjective of this form in EDICT is 執念い (しゅうねい), which I had never heard of. The pattern above is estimated from the discussion below.)
These forms of ウ音便 are the result of two sound changes.
- In the first step, く was replaced by う: はやく→はやう, たのしく→たのしう, さむく→さむう, おそく→おそう.
- In the second step, the -au became -ou and the -iu became -yuu: はやう→はよう, たのしう→たのしゅう. (This sound change is not limited to ウ音便 of i-adjectives, and in other words, -eu became -you with pronunciation /joː/. This is why I wrote the ウ音便 of i-adjectives ending with -ei should be -you.)
Why sometimes with the prefix お and sometimes without お?
I can explain this only partly.
The prefix お shows respect. おはようございます literally means “You are early,” and おめでとうございます literally means “(what happened to you) calls for celebration.” In these cases, the adjectives describe something about “you,” hence the prefix お. On the other hand, in the sentence たのしゅうございます “I have fun,” the adjective たのしい describes the state of “me,” hence no prefix お.
However, this argument suggests ありがとうございます should have the prefix お because it literally means “What you did is hard to exist,” and the subject of ありがたい is the action of “you.” I do not know any explanation why ありがとうございます is not おありがとうございます.