Q1: Beam weapon accuracy, broadly, is determined by:
Accuracy = [ ( 1 - target_range / BFC_max_range) * ( missile_bonus * BFC_speed / target_speed ) - 0.10 * (enemy_ECM - self_ECCM) ] * weapon_accuracy
A couple of the terms here of course cap at 1 or 0, but I've omitted that for readability. "Missile_bonus" is any benefit you get from the missile tracking time tech, and weapon accuracy is usually 1.0 for most weapons but reduced-size Gauss will have fractional values. Note that aside from this last term, accuracy is determined entirely by the BFC and not by the weapon (even the weapon range).
Q2: There are two differences between Gauss and Railguns to be aware of. One as you've noted is the number of shots, the other however is the fact that Gauss can be turreted which allows it to use your maximum BFC speed (racial tracking speed * 4x size). A typical ion era railgun ship might have a racial tracking speed of 4000 km/s, thus a theoretical maximum of 16,000 km/s, however if the ships only go at 5000 km/s then this is the maximum tracking speed the railguns can use for PD. You can instead put them into Gauss turrets using the full 16,000 km/s and they will be more effective on a per-weapon basis.
It depends somewhat on your turret, engine, and fire control techs, but generally the consensus is that Gauss breaks roughly even with Railguns at Gauss ROF 4, and becomes a superior PD weapon at ROF 5 and higher. The flipside is that by the time you have ROF 5+ (which is a 30k RP tech), your PD railguns while not as effective are usually able to be strong dual-purpose weapons while Gauss is pretty crap at anything that isn't PD. Thus either is fine.
On the topic of PD it is also worth pointing out that lasers and mesons can also be mounted in turrets which makes them passable as PD weapons. For mesons this is not important because they suck, but if you start with laser techs they can make passable PD weapons with a lot of dual-purpose capacity, although they will never be as good at pure PD as Gauss or Railguns. Depending on what kind of fleets you're up against, laser PD turrets may be "good enough" allowing you to save RP by not investing into Gauss.
Additionally, a mature anti-missile defense strategy should include AMMs if research permits as these can help weaken large salvos before they reach and overwhelm your beam PD.
Q3: Weapon damage for lasers, railguns,
and plasma (correction to above comment) falls off linearly with range.
Damage = max_damage * (1 - target_range / max_weapon_range)
Other beam weapons have no falloff due to dealing one damage (Gauss, meson, microwave) or being particle beams.
You also want to make sure you are aware of the
damage profiles for different weapon types. Plasma and missiles use gradient (1), particle beams and railguns use gradient (2), and lasers use gradient (3). Particle lances deal damage in a single armor column, and the other weapon types do one point of damage and have no gradient.
