Part 1:maille This really depends on the date.

When plate armour was first introduced as a pair (coat) of plates and supplementary limb defenses, it was worn over a full maille shirt. BY the middle of the 14th century maille sleaves and a skirt and sometimes a maille collar could replace the maille shirt under the pair of plates. Inventories of the 14th century do not record many chauses (maille hose), so it appears these fell out of use early. As the century progresses and plate armour becomes more complete, maille gussets for the underarm and underside of the upper arm/elbow become more common, and seem to replace sleeves for full armours over the course of the 15th century The maille collar, or standard, is replaced on full armours by a plate gorget around the turn of the 15th/16th century. The maille skirt remains in use through the 15th century, though sometimes maille shorts of a kind were used instead. You see fewer skirts in art in the 16th century, possibly because they were not necessary when articulated tassets join the fauld (skirt) and cuises (thigh armour). Some form of supplemental maille armour stays in use through much of the 16th century at least.

EDIT: It should be noted that the maille shirt didn't disappear as an under-layer after 1400, and maille sleeves didn't disappear either. The supplemental defences continue to be worn, sometimes depending the the configuration of armour. Anthonio Moro painted Philip II in what was at least maille sleeves and a skirt (and possibly a full shirt) under a breastplate in 1560.

Generally plate armor would be worn over a shirt known as an arming coat or an aketon. The exact construction could vary but they would usually be made out of padded leather with laces at various points to attach the armor. Many designs would incorporate pieces of mail to reinforce areas that would be more exposed such as the armpit or the inside of the elbow, for instance this example.

I could have a go at answering this question but I think /u/WARitter is probably more qualified to do so. He's also currently reading Claude Blair's European Armour c.1066-1700 so this type of stuff should be more readily at his fingertips than it is at mine.

Depends on the area and time - for example, 16th century Italian plate did indeed have small sections of mail covering areas like the armpits and the crook of the elbow, but not a full hauberk. I'll have to look for my sources, but in the mean time, could you be more specific?

EDIT: Okay, as a source to my claim that small segments of mail were used to cover unprotected areas: in 2013, the Metropolitan Museum of Art posted This video showing late 15th century full Italian Plate Armor, its assembly on the wearer, as well as fabulous narration from the assistant curator.
As you noted below, there were certainly different styles, methods and types of plate armor. In truth, plate armor began much earlier than the 14th century. (SMALL ASIDE:) Plate armor has been used through most of recorded history in some form or another. The Greeks used some metal breastplates, and the Romans famously used segmented plate armor (lorica segmentata). Mail was really used for a (comparatively) small period of time, beginning, if I understand it correctly, with the Celts in about 500-300 BCE. Largely, the type of armor used was largely dependent on the enemy one was fighting against along with the resources available (with some exceptions). Plate armor is continued today in modern armors mixed with Kevlar, rather than mail, and is critical in protection for solders, police and corrections officers alike. (BACK TO YOUR SPECIFIC QUESTION:) In the 15th and 16th century, plate armor included various amounts of mail in it, from a full hauberk underneath a breastplate and to just small segments (as observable in the linked video). While I am not aware of any examples, and I doubt the existence of said examples, I suppose it is possible that SOME plate armor was intended for use without mail: practically speaking, armor of that design would have been prohibitively costly to create, and terribly uncomfortable to wear, particularly in the armpits and groin.