1- I read this somewhere: For He exists with the attribute of perfection, and the fact that He can be seen both by Himself and others is among the characteristics of perfection. Does this mean that Allah can see Himself?
2- Prophet Mohammad, sallallaahu 'alayhi wa sallam, said, "If one of you lowered a bucket by a rope (into a well), then it would fall on Allah." [At-Tirmidhi] What does this mean?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

Ahl As-Sunnah are of the view that Allah sees Himself, as mentioned by Shaykh Ibn Taymiyyah in his theses and in his responses to the Jahmiyyah sect, and he differed in opinion from some misguided sects, who claimed that Allah Almighty does not see Himself and is not seen by others.

Al-Ash'ari said in Al-Ibaanah ‘an Usool Ad-Diyaanah, in response to them (p. 53):

"What proves that Allah Almighty is seen by sights is that Allah sees things. Since He sees things, then the one who sees things can also see Himself. Since He sees Himself, it is possible that He makes us see Him. This is because the one who does not know himself does not know things, so since Allah knows all things, He knows Himself. Likewise, the one who does not see himself does not see things, so since Allah Almighty sees things, He also sees Himself. Since He sees Himself, it is possible that He makes us see Him. This is the same as saying that since He knows Himself, it is possible that He makes us know Him…"

As regards the hadeeth, Shaykh Ibn Taymiyyah classified it as weak, and so did Shaykh Al-Albaani.

Shaykh Ibn Taymiyyah clarified its meaning as he stated in Majmoo’at ar-Rasaa’il wal-Masaa'il:

"The hadeeth of lowering the bucket was narrated by At-Tirmithi and others from the hadeeth by Al-Hasan on the authority of Abu Hurayrah, which is an interrupted chain of narrators, because Al-Hasan did not hear from Abu Hurayrah, but the Marfoo‘ (attributed to the Prophet)  hadeeth of Abu Tharr strengthens it. So if this latter hadeeth is confirmed and authentic, then its meaning corresponds to this one. His saying, 'If one of you lowered a bucket by a rope (into a well), then it would fall on Allah,' is a hypothetical consideration; that is, if the lowering happened, then it would fall on Allah. It is not possible for anyone to lower anything on Allah, however, because Allah's self is high, and if anything was lowered in the direction of the Earth, then it would stop at the center, and it would not go up in the opposite direction (from there). However, if there was a hypothesized lowering, then what he (the Prophet) said would happen (because Allah, the Exalted, surrounds all things)."

Allah knows best.