Every year there are umpteen IMGs who have the question if it is correct to waive or not. This is including me, as until I started working on the letters of recommendation the word waive was relatively unknown. I remember, I even had to go to the Webster dictionary and thesaurus to make sure that I understand the meaning of the word before I assigned my life's crucial decision. The best-case scenario is when the office at ERAS receives the letter directly from the referee (person writing the letter) and you have waived the right to see the letter.
ERAS will make no distinction, when the LOR is sent to them. They are just a middleman who is processing your application. The difference comes when the letters are scanned and then sent to the respective programs. There are four possible scenarios to consider:
letter sent directly - meaning the referee mails the letter to ERAS
Indirectly - meaning the letter gets back in your possession and you mail it in.
1. Letter sent directly to ERAS - waived option checked = considered to be waived, contents highly regarded as unbiased and valued.
2. Letter sent directly - not waived = considered to be not waived, considered to be biased, less than ideal situation.
3. Letter sent indirectly - waived option checked = considered to be NOT waived, as there is no way for ERAS or program to ascertain that the secrecy of the letter is maintained.
4. Letter sent indirectly - not waived = considered to be not waived, biased. Least favorable scenario.
So you can see, if you were in possession of your letter, then there is no way for ERAS to know if you are aware of the contents of the letter and thus is considered to be not waived. SO rather than take the letters in your possession in good faith given to you by the faculty, please direct their office to send it to the ERAS office along with the LOR cover letter submission form.
You can send more than four letters but it is not necessary as it will NOT be more advantageous. Think of it as law of diminishing returns. Chose the referee who YOU think is going to be the BEST SPOKESMAN for you. In general, US LORS are better recognized than foreign LORs because of familiarity with the system. Titles etc. in your case better to have a good mix of both.