this is about the timing structure. I'll use Data Raven to clarify my question(s).
Let's say I am the corporation and I have a Data Raven with a power marker (yeah!!!). All the beginning of turn stuff has already happend and I enter the action phase:
1. I sacrifice the power token on Data Raven to mark the Runner
--> The ability on Data Raven doesn't require a click. Is this considered an action?
Correct! It is not considered an action, as it does not use a click.
--> Can the runner now use abilities or act in any other way?
In Netrunner there are "Paid Ability Windows" in which all non-click abilities can be used. Check out the timing chart in the back of the manual for all the different points where these timing windows occur.
The way the timing window works is that the player who's turn it is (in your example, the corp player) has priority. They can do all non-click paid abilities that they wish, all at once and in the order they choose. Once they have completed all their paid abilities and effects, their opponent gets priority and can do all their non-click paid abilities that they want. When they are done, priority passes back to the original player and so on until one player passes (does not use a paid ability).
So, in your example, once you have spent your Data Raven token to give the Runner a Tag and are finished with paid abilities the Runner then gets an opportunity to use paid abilities.
2. I play Scorched Earth for four meat damage.
--> This requires a click and should be an action.
--> Can the runner now act or can I just go to the next action?
There is a paid ability window between each click. Refer to the timing chart.
--> If the runner could act, what are ways to get rid of markers during the con's turn? Are there any?
I assume by "markers" you mean "Tags". There are very few ways that Runners can get rid of tags on the Corp turn. The console "Forger" (from the "The Underway" datapack) allows the runner to trash to remove a tag, for instance. The runner does have a number of tools to avoid taking tags in the first place. New Angeles City Hall (From the "Future Proof" datapack), for example, allows the runner to pay two credits to avoid taking a tag. "Avoid" abilities are used right as the thing they are avoiding happens though, they are not used as a separate paid ability.
3. I use my remaining two clicks for private army and cause two more damage.
--> Same question: Can the runner act in between? What could he do?
Yes, there is a paid ability window between each click. There are some cards that allow the runner to prevent Meat Damage (for example, Crash Space) and some that allow the runner to draw cards (for example, Sports Hopper from the "Salsette Island" datapack). Prevention abilities are like the avoid abilities, they are used right as the thing they are preventing happens. Other than those two things though, if the Runner is in this predicament they're in trouble .
edit: formatting out ninja italicizing