Usage of the word "appealing"

I find appealing the recent methods that he has been using to grow his business.

The usage of 'appealing' (the adjective form) is correct here, however you also use 'appealed' (the verb form). But the sentence is unidiomatic, as well as, poorly structured. You might consider using any one of the following alternatives:

• I find the recent methods that he has been using to grow his business appealing.

• The recent methods (that) he has been using to grow his business is appealing to me.

• The recent methods (that) he has been using to grow his business appeals to me.

• The recent methods (that) he has been using to grow his business appealed to me.

I am appealed to Dr. Smith's research on biology.

On similar terms, 'appealed' is correct here. The sentence is grammatical. You could also go for:

• Dr. Smith's research on biology appeals to me.

• Dr. Smith's research on biology is appealing to me.

• Dr. Smith's research on biology appealed to me.

Although, any of them can be used, you must keep in mind to alter the tenses correctly to fit the context.