Touching the Wall: Kotel Konnections
Chazal tell us that the Shechina never left the Western wall of the Beis HaMikdash (Shemos Rabba 2:2). The Midrash clarifies that the reason Hashem can never destroy the Western wall because it is the place of the Shechina (Eicha Rabbasi 1:32). The Radvaz (Shut Radvaz II:648) explains that this does not mean that true Kedusha remains only at the Kotel, rather the Makom Kodesh HaKodoshim remains Kadosh even after destruction. The addition is that the Makom Mikdash is most potent in the West. However, one davening on Har HaBayis (when he has no choice to be there) does not face the Kotel during Tefilla. He faces Makom HaKodesh HaKodoshim.
In fact, there has been a long standing tradition in Judaism, stretching back to the times of Yaakov Aveinu, to pour our hearts out at the Kotel (Pirkei D'r'eliezer 35:Rashi <Berashis 28:19>). The Gemara (Berachos 31a) suggests that the sanctity of the place adds to the Hislahavus of the one who davens there. But can one benefit from the Kotel? May one kiss the wall or take part of the stones home?
The Radvaz (II:691) notes that in his opinion, the wall of the Kotel is, in fact a wall of the actual inner sanctum (Azara) of the Beis HaMikdash. Given his position that the Dome of the Rock is the Makom Kodesh HaKodoshim, it would follow that the Kotel of today is the wall of the Azara and therefore, a Baal Keri and a Nidda (even a girl prior to her wedding) may not approach the Kotel (See Chochmas Adam, Mishpitei HaAretz, 8). According to this position, not only does one not have permission to touch the Kotel, many might not even be able to approach the Kotel. Beis Radvaz (38) and Shut Har Zvi (Yoreh Deah p. 269) try to explain Minhag Yisrael despite the position of the Radvaz.
However, it has been noted that the wall of the Kotel is not really the wall of the Azara. The Avnei Nezer (Yoreh Deah, 450) clearly notes that the Kotel must be the wall of Har HaBayis and not the Azara. Rav Yechiel Michel Tuckechinski (Sefer Eretz Yisrael) notes that in the days of the Kaftor V"ferach, the Kotel was not even able to be seen. Accordingly, he feels that it is clear that the Kotel is not inside the Har HaBayis but outside it. If it had been part of Har HaBayis, it would have never been covered (it would have been on top of the mountain). There are those who are still Machmir not to visit the Kotel for this reason (See HaMiasef, III:121). The Griz was included among them (See Iggros Moshe, Orach Chaim II:113).
A question was put forth to Maharil Diskin as to whether people may put notes into the Kotel wall. The Maharil Diskin (Sefer Mishkenos L"Avir Yaakov) reportedly did not feel that he needed to deal with the question given that the Kotel did not have Kedushat HaMikdash according to him. The Aderet too, felt that one may put his fingers against the wall of the Kotel for it is not Kadosh. However, if it is not Kadosh, why do somany choose to specify the beauty of Davening there?
Rav Yitzchak Yosef (Yalkut Yosef Dinei Har HaBayis) cites a Taz who feels that the Kedusha of the Kotel is that of being the largest Shul (Siman 151). Thus, even if we adopt the prevalent assumption that the Kotel is not the wall of the Azara, it is an especially significant Makom Kadosh. The only Chiddush is that according to the view of the Taz, one might recognize a Kedusha of the Kotel without seeing the area as Meila (see Har Zvi Yoreh Deah, p. 281).
So how far does Kedushat HaKotel extend today? Rav Moshe (Iggros Moshe, Y.D. IV:63) felt that although we do not see the area of the wall as Kotel Azara, there is still Meila if one takes part of the stones. This is based upon the idea that the Kedusha of Kotel Har HaBayis was never removed from the original wall. The Tzitz Eliezer (x:1) and Rav Ovadiah (cited in Yalkut Yosef vol. II) agree. Still, the prevailing opinion permits one to place Kvitlach into the wall as the outside and the spaces in the stones were not Mikudash (see Avnei Nezer above). In cases of great need, Rav Ovadiah allows the bushes in the Kotel to be trimmed. All agree that the Kedusha of the Kotel is clear and needs to be respected.