1- Most historians state he went two times while some historians state he went three times to Yemen.
2- The first was year 8 while the second was year 10. Those who add a third state it occurred between year 8 an 9.
3- Yes you are right. The objecting and complaining took place only during the first one (year 8) not he second one (year 10). Sunnis have conflated these two to undermine Ghadeer.
If you read Arabic, you may read this short article:
https://www.aqaed.com/faq/1095/
And also this part of the book اعيان الشيعة which is more detailed:
http://shiaonlinelibrary.com/الكتب/3636_أعيان-الشيعة-السيد-محسن-الأمين-ج-١/الصفحة_405
And you can read some more details here about the first trip:
https://www.aqaed.com/shialib/sia06/6.html