To add or not to add adjective after 一樣。
Yes, 高 is superfluous. There is no need to say two 高 in a sentence.

I'd rather say 我跟他一樣高. Although you can also say 我跟他身高一樣, the former one is a little bit shorter.

If you say 我跟他一樣大, one might not know what kind of '大' it is. In this case, one might add 年齡 to the sentence. But if the context is known in conversation, it is no need to add 年齡 to the sentence.