Please explain MOSFET output characteristics and load line when used as a switch
In what region should MOSFET operate in &quot;on&quot; state? Linear or
saturation?

For this I've assumed the load current is 30 mA because it matched with a graph of MOSFET characteristic I had close to hand. For your load of 90 mA the green line will cross near the Y axis at 90 mA if you can supply enough \$V_{GS}\$ voltage: -

When &quot;on&quot; the MOSFET will be in the linear or ohmic region. When &quot;off&quot; the MOSFET will be in a part of the saturation region called &quot;cut-off&quot; i.e. right along the curve for when \$V_{GS}\$ is at 0 volts. Current leaking through will be around 1 μA.

What is the point of the load line in the output characteristic?

For your application there is no point at all - you have two points; one close to the Y axis and one close to the X-axis providing you can supply enough \$V_{GS}\$ voltage: -

But your \$V_{GS}\$ voltage is only 3.3 volts hence it will look more like this: -

And, for the characteristic picture I chose, both ends of the load line will be in the saturation region.

How do I find a MOSFET with a low enough threshold voltage to turn
fully on at 3,3V?

You have to dig a little and possibly use website comparison pages like those produced by Farnell, Mouser and Digikey to name but three. You are probably going to narrow down your search by selecting the maximum voltage (\$V_{DS}\$) to be no more than 30 volts and you will probably find that selecting a device with maximum drain current of at least 5 amps is going to put you in the right area for picking a device.