1) A pillar was a solid column of coal, left unwrought in order to support the roof of a working.
It is first recorded in a case of encroachment brought to the attention of the Chancellor of the Duchy of Lancaster by Henry Farrer: 1591 John Drake and John Roebucke have now of late at several times in most forcible manner, felled and cut down all the heads, pillars and other work being placed and made within the grounds of your orator’s mine at his great charges for bearing up the ground there, Northowram. More usually it is found in leases, included in the list of conditions under which the coal-mine might be worked: 1659 doth … agree … that he will at all tymes … durynge the said term keep and maintaine in all such pitt or pittes … good and sufficient pillers for the upholding and supporting of the groundfeild … as is used and accustomed, Wibsey
1777 and leave the Pillars thereof of a sufficient size and strength, Southowram. Such pillars might at a later stage be exploited, especially if at ground level there was no important building, and that is implicit in the Sale of pillars of coal left unwrought in Beeston in 1787.
2) No definition available
Although this is in regular use for the pier of a bridge, the first OED evidence is in a Nottingham document of 1579 - relatively late. The new stone bridge at Catterick in 1422 was to have twa pilars and Elland Bridge two substantial juells or pillors in 1579 but the word is seldom used after that. I have found no examples locally of ‘pier’ although its history in this sense goes back to the twelfth century in Kent.