(b) They went to the Botanical Garden, where there were some/many rare plants. They took some photos in the garden. -- "also" means "in addition to something else", but you haven't mentioned anything else. "There had..." is ungrammatical in this sense. It's possible to "capture" a photo, but it doesn't seem natural here. Usually you would use "captured" if the photo was particularly precious or difficult to obtain.

(c) The Botanical Garden had many plants from Japan. -- OK

(d) They went into the garden. They were interested to see the plants.

The use of "the" before proper names (capitalised words) is complicated and idiomatic. In fact, there hardly seems any logic to it at all. Often you just have to memorise individual patterns.

In this case, a plausible explanation is that "botanical" and "garden" are just ordinary descriptive English words. You could write "They went to the botanical garden", which requires "the". When you capitalise the words to show that it's considered a proper name, you still need "the". (In fact, it's debatable whether you should capitalise "Botanical Garden" at all, but in my view it's OK if it's a notable place.) This "rule" will not always work though.

"They were interested to see the plants."

"the" is required here because you're referring to some specific plants (the ones at the botanical gardens) rather than plants in a general sense.