Welcome to our Forum and thanks for your questions.  I'll be glad to help.  I'll go straight to your questions:

1.  There is no meaningful risk in having your blood drawn for testing in a licensed, regulated commercial laboratory.  The have detailed procedures and take many steps to avoid even accidental needle reuse.  In most settings using vacuum tube to draw blood. the devices used to get the blood from your vein into the tube are designed to only be used a single time.  In settings where syringes are used, again. the syringes are typically disposable and only used a single time.  In a well established commercial lab there should be no concern about having had your blood taken using a used needle.  Thus there would be no risk for infection of any sort.

2.  HIV is transmitted only through DIRECT, unprotected sexual contact with an untreated infected person.  It is not acquired through the usual activities of daily living.  I would have no concerns that you would have been infected since the last sexual encounter that you describe above.

3.  From what you have said, your 31 May test results should be reliable and there is no need for further testing.

I hope this information is helpful.  I am confident that when you are tested for your visa the test will should that you were not infected.  Thus I see no reason for concern or for further testing.  Please feel free to use your up to two follow-up questions if any part of this reply is unclear.  EWH