The weapon info screen shows 3 slots. They seem to be for the unique upgrades of each gun. Am I limited to only 3 upgrade installations per gun or can I add for example silencer, AP, laser as well as damage?
Take a look at this in-game image of the 10 mm pistol (taken from Wikia):
The three icons next to the weapon image always represent (in order) the Laser Sight, Silencer and a Specialist upgrade. If a weapon cannot take any of these specific upgrades they will be crossed out and they would be highlighted green if fitted. We can see in this example that the pistol can take all three, but has none yet fitted.
Below these are the weapon's four statistics. The bright orange elements here show the weapon's base performance, then there sometimes dark green bits following these. The number of dark green bits in each line is how many weapon mods of that type the weapon can take, and these become green when fitted. If there are no dark green bits the weapon cannot take any upgrades of that type.
Continuing on from the same example, let's look closer at this pistol's damage:
Base damage is three units, one damage upgrade has been fitted, and the weapon will take another three (for a total of four damage upgrades).
What this means in this example, the 10mm Pistol, can take all of:
For a maximum total of 19 different upgrades of varying types.
No, you are not limited to three upgrades per gun. The three upgrade slots you see, are actually for attachments and unique upgrades, like silencer, laser sight, armor piercing, scope, etc.
The damage output, reload speed, ammo capacity and rate of fire upgrades are different from the attachments and unique upgrades. While examining a weapon, you can see how many times to can upgrade an attribute by looking at the green bars next to the specific attribute.