I hope you could answer some of my questions this time

Question

Can you say: I use the Internet to do my homework researches or is it better to say for doing my homework research/for homowork research?

After a little period OR after a short time OR after a while ???

In a first time OR in the beginning he enjoyed being Hyde???

M.L.King had a dream that all human races would have the same rights OR DREAMED THAT oR only wished is possible???
Alternatives to animal testing are analysing statistics OR statistical analyses???

Thank you

Answer 1

Can you say: I use the Internet to do my homework researches or is it better to say for doing my homework research/for homowork research?
I'd say, "I use the Internet to do my homework and research."

After a little period OR after a short time OR after a while ???
It would be good to have a whole sentence. Any of these would do, but it depends on the rest of the sentence. What did you have in mind?

In a first time OR in the beginning he enjoyed being Hyde???
"At first, he enjoyed being Hyde."

M.L.King had a dream that all human races would have the same rights OR DREAMED THAT oR only wished is possible???
I'd say, "M. L. King, Jr., had a dream that all humans would have the same rights."

Alternatives to animal testing are analysing statistics OR statistical analyses???
I'd say, "Alternatives to animal testing are analyzing statistics." (Or you would write "analysing" if you are in England.)

Answer 2

OR -- for that last one:

"Alternatives to animal testing include the analysis of statistics."