Is there a problem with wearing sandals or Crocs during Davening instead of regular footwear? What is the source for this issue and what is the reason for this issue?
The halachic gauge for how to dress for Hashem is how people dress in front of "great people" and every place is different (Shulchan Aruch O.C. 91:5 MB 12-13). I assume that Israel has a more laid back standard than, say, the U.S. In Israel, people might visit the Head of State in sandalim.
This opens a big can of worms. Can a person wear jeans and a polo shirt? Unpressed slacks? No tie? Scuffed shoes?
This depends on the location. R Shlomo Aviner writes in his book On the air (full book available here), p. 40, that it is acceptable in locations where wearing sandals (he speaks without socks) is respectable.
One is required to daven dressed in a way in which he would stand before a king. How one stands before a king is dependent on the place and time in which one lives. The Beit Yosef says that one is obligated to daven barefoot (Orach Chaim 91:5). Not only is it permissible but one is obligated to daven barefoot. Obviously, today if someone shows up barefoot to daven, he would be thrown out of the shul.
Therefore, it is dependent on reality. Wearing sandals nowadays is respectable (this applies in Israel; other locations may differ). Wearing flip-flops, however, is not considered respectable since a person wears them when getting out of the shower or at the pool or on the beach.