Can I use my lecture materials on other platforms like Youtube?

You may upload your lecture materials on other platforms.

However, please be noted that you should not submit the same materials to other institution for any other subsidies.

How can I upload files to google drive when my storage is full?

 

If I give my material away openly, what happens if someone takes it and claims it as his/her own?

You are not putting your material in the public domain. What you are doing is applying an open license that reserves some rights for you. Users cannot take your material and claim it as his/her own because the license mandates that the user properly attributes the original author.