Do the three pillars have any religious significance other than structural support for the roof?
Well to be honest my answer will be partly opinion based: As i could exclude a religious significance for the 3 pillars (as AFAIK no body exactly knows if the 3 of them were there at first or have been built in later, as ahadith only mention the two which are in direction of Yemen others mentioned that it had 6 pillars) i think they have only an architectural/statically significance. Because the significant religious parts of the Kaaba are more or less outside: the black stone, maqaam Ibrahim (peace be upon him) and the Kaaba cube itself as the Qibla (direction for the prayers).
Who is allowed to enter inside?
You should know that the Kaaba has been built and re-built several times and after the time the sahabi Abdullah ibn az-Zubair (may Allah be pleased with him) has re-built it according to a Hadith narrated by um al mu'minyn 'Aisha (May Allah be pleased with her) see here or here which shows that the Kaaba had 2 doors and even according to some ahadith and narrations by scholars of the early 3 centuries of Islam one could enter it and pray (nafl) or make dua' in it. Now apparently it is closed and only the Family of Bani Shaiba بني شيبه an Arabic tribe (from Quraysh) holds the keys to the Kaaba according to some ahadith or narrations as they are descendants of 'Othman ibn Talha (May Allah be pleased with him) who hold the keys when Mohammed (peace be upon him) entered Mecca in year 8 AH.
Here some other references for worshiping inside the Kaaba:
From Sahih al-Bukahri,
in Riyyad as-Salihin,
from Sahih Muslim 1 2,
from Sunan an-Nasa'i 1 2 3 (the 2nd hadith narrated by ibn Abbas (May Allah be pleased with both) is a kind of counterexample)
and Jami' at-Tirmidhi (in the Arabic comment you can read that Imam Malik ibn Anas said that there's no harm in praying inside the kaaba for nafl and he said it's makruh to pray a fard prayer inside, while Imam a-Shafi'i said there's no harm in praying nafl or fard inside the kaaba as they have the same conditions of purity and qibla). So apparently the Kaaba was still open(ed) around 200 AH!
And this would mean any body (of course i mean Muslims as non-Muslims are forbidden to enter the Haram) could enter it. But as now the doors are kept closed expect for the cleaning ceremony (AFAIK) nowadays this won't be easy anymore!
Further informations about the rebuilding of the Kaaba could be found here and here and in Wikipedia of course!
And Allah knows best!