I think 1 is in the FAQ.

I've improved C+ to A+ at D3 by appearing in NT Champ game

Totally dependent on how your team does.  In theory two letter grades as above.

Hugely.  Less so for battles at D3, but you will see better recruits than someone with a lower prestige.  What may be a pulldown for a c+ prestige may show up as directly at your level with a a- (for example, the numbers are rough, but theory holds)

off top of my head, prestige is determined by:
W/L record (I think there is a mild bump for winning record)
Post season success in NT and PI, as well as going to post season often
Conference championships and Conference tourney championships
In D1 - I think lots of players getting drafted helps

I haven't kept track of typical improvements in one season. I would guess 1 letter grade is very good, often 1/2 letter grade will happen. You could look at successful teams under the history tab and get an idea of the progression.

I think a new player can expect a few seasons (4-6) to be to B+ prestige, if starting at C-... it really depends on how quick you learn the game. It took me awhile with my first D3 team to learn and understand things.... but after I learned some things I was able to grow D3 programs to respectable level pretty quickly (3-4 seasons).

Prestige, at the D3 level, helps with getting first dibs on recruits. higher prestige teams will have more players listed on their D3 board, and as D2 players drop down they will hear about it first. Once a battle is engaged - prestige does not matter at D2 and D3 from what I gather. At D3 the determining factor in winning battles is often how much $$ each school has (which you can figure out) and distance from the recruit. Prestige often gives the best schools a head start on players, but it may not matter if the player is closer to the lower prestige school. (recruits >360 miles away are significantly more costly to do home visits, etc.).

At D1 it is very important. For instance, an A+ prestige school at D1 may only have to expend 1/2 the effort on a recruit to win the battle vs. a C+ school....

At D2 and D3, prestige is based on only two factors:

50% is your current prestige;
50% is your current season's results. 

So, if your current prestige is C+ and you win the NT (A+), your new prestige rating should be B+

We don't know  the criteria for an  A+ season vs. an A season or an A- season.  But the  number of wins can be just as important as your post-season results. 

At DII,  I started out 1-26 and was a D+ prestige.
The next seaon, I was 14-14 and improved to C-
After a 26-4 season, losing in the first round, I was B-
After 31-3 and making the Final Four, I'm now A

Posted by alblack56 on 7/24/2014 3:42:00 PM (view original):
At D2 and D3, prestige is based on only two factors:

50% is your current prestige;
50% is your current season's results. 

So, if your current prestige is C+ and you win the NT (A+), your new prestige rating should be B+

We don't know  the criteria for an  A+ season vs. an A season or an A- season.  But the  number of wins can be just as important as your post-season results. 

I don't think the 4th line is accurate...take a look at my old UMPI team (seasons 62 and 63).  In D3, every time I've seen a team make the FF or better their prestige has gone to A+.  I realize a lot of those teams already had good prestiges to begin with and that's why I'm still trying to find a better example.  But B- to A+ in 1 season in D3 is definitely doable.  

It's definitely not true.  I took over Piedmont, ran it into the ground a bit, then making championship game season 90 took me from a C+ to an A+.  Maybe it went from a high C+ to a low A+, but two letter grade jumps can absolutely happen. 

Taking a D+ to a B- in four seasons at a low D1 school is I'm sure much more common, but here's an example at New Hampshire.