@Trialben covered most of it properly.
To fill your keg from the fermenter, it is preferable to run a hose from the fermenter to the keg "out" port: It has a long dip tube that puts the beer at the bottom (as opposed to the "in" (gas) port, with its short dip tube that'll let the beer drop several inches and splash/oxidize). It is not preferred to pour the beer in through an open lid, but do what you have to. Gravity feed is OK.
Ideally you will have your keg clean and sanitized, with the lid in place. Purge the O2 as directed in the instructions you posted. Then connect up the beer 'input' hose to the "out" port (unscrew the tap), and a 'blank' quick disconnect to the "in" port (the pic you posted shows it 'blank', meaning not connected to a regulator or hose, so shut off and disconnect the regulator) to allow pressure to escape. You stop filling when liquid comes out this port (assumes you have more than 1.75 gallons of beer). There is some gas head space at the lid, so if the lid is in place you cannot overfill it. You do not need a lot of 'head space' but some is needed.
If you pour it in, then leave an inch or so, or more. 1.3 gallons easily fits in a 1.75 keg, so you'll be good there.
To carbonate it, use a carbonation chart, which shows CO2 volumes by temperature and pressure. A really fizzy Hefeweitzen might want 4 volumes, a typical ale 2.5 volumes. Chill the keg good and cold (I use 38F) and first pump in maybe 30 to 40 PSI. Give the keg a shake, and you will hear gas flow in. Repeat a few times, then let it sit overnight to finish letting CO2 dissolve in the liquid. Re-set the pressure the next day to what the carbonation chart says. (You can start at the carb chart value, but it can take a few days to a week to fully carbonate).
Drop the pressure to maybe 5-8 PSI for serving. For storage, I usually bump the pressure back up to the carb chart value, or it tends to go flat.
A 74 ounce tank will last a few kegs worth. You should have a shutoff after the regulator, as it generally will not go down completely to zero PSI. A hardware store or Harbor Freight valve will do, $6. It can wait though, because you can leave the CO2 connected most of the time. But you don't have to!
Note that condensation in a regulator - from chilling it with the keg and then bringing it out to the normal atmosphere - will eventually cause damage. Try to avoid that.
Lagering in the keg: Yes, no problem, but I suggest you only give it a few PSI (3-6) with the goal of keeping O2 away from the beer. Lagering is a mechanical process, in that the 'haze stuff' sinks to the bottom of the keg. The dip tube will dispense that with the first pour, so expect that the first few may be cloudy.
!
