It's a surprisingly complicated question. Given your mention of friction, probably the main point is that for a car tyre the friction is not linearly dependant on load. Wikipedia has some information about this here.
If you had perfectly smooth surfaces the friction is actually proportional to the area of contact and independant of the load. This is because friction is an adhesive effect between atoms/molecules on the surfaces that are in contact. However in the real world surfaces are not smooth. If you touch two metal surfaces together the contact is between high spots on the two surfaces so the area that is in contact is much less than than the apparent area of contact. If you increase the load you deform these high spots and broaden them, so the effect of load is to increase the real area of contact. The real area of contact is approximately proportional to the load, and the friction is proportional to the area of contact, so the friction ends up being approximately proportional to the load.
However a rubber type is a lot softer than metal, and a road is a lot rougher than a metal plate. Even at low loads the tyre deforms to key into the irregularities in the road, so increasing the load has a lesser effect. That's why you get the sub-linear dependance described in the Wikipedia article.
But this is only the start of the complexity. If you use a wider tyre the contact patch area isn't necessarily bigger. A wider tyre has a wider shorter contact patch while a narrow tyre has a narrower longer contact patch. The contact patch area depends on the tyre pressure, the deformation of the sidewalls and probably lots of other things I can't think of at the moment.
And anyway, if by "grip" you mean grip when cornering, the grip isn't just controlled by the contact patch area. When a car is cornering the contact patch is being twisted. This is known as the slip angle. The wider shorter contact patch on a wide tyre has a smaller slip angle and as a result grips better.
No. Tread is there to account for the presence of "other material" between the tire and the road surface. For example, if there is water on the road, a perfectly smooth tire would be susceptible to aquaplaning: the water has nowhere to go, and becomes a thin film between the tire and the road.
If anything, deep profile tends to reduce the area of the tire in contact with the road, and increases the pressure. Both of these things give the "other stuff" (water, snow, loose gravel etc) a chance to get away from between the tire and the road, allowing the tire itself to grip the road.
In situations where there is a thin layer of snow (and to a lesser extent, mud or clay), a deep tire grip can shape the snow into a waffle like pattern that interlocks with the tires, and results in (much) greater lateral force - no "simple friction", but the kind of resistance to relative motion that comes about from having interlocking structures (like the teeth of gears). The deeper and wider the profile, the better the chance of a solid ridge of snow/mud being built up.
\Once you have the "intended" interface between tire and road (without the other material), the force of friction is largely independent of area, since the total normal force will be constant. But most of the time, tires slip because you are not dealing with just the road; and the profile is intended to deal with "the other stuff".
Best Answer
Lower pressure increases surface contact and increases static friction, and static friction does not involve heat loss, so that is good. But rolling friction is not good and does involve heat loss.
Rolling friction heating is due to the inelastic deformation the rubber of the tire experiences when it is in contact with the road. See this article on rolling resistance from Wikipedia: https://en.wikipedia.org/wiki/Rolling_resistance
When the rubber is in contact with the road for each revolution, it is compressed, it then expands when it leaves the surface. The compression and expansion is not perfectly elastic, thus there is heat loss in the form of friction. The lower the tire pressure, the more rubber that is in contact with the road for each revolution, and the greater the friction heat loss. These increased heat losses add up to lower fuel economy.
Hope this helps.