I know that they have lift pumps that bring the fuel towards the plungers so that the plungers can push and pressurize the fuel towards the injectors but i was wondering if lets say we removed the lift pump, would the mechanical plunger injection pump be able to pull through the diesel fuel from the fuel tank to fill in the plunger area in the injection pump and then push the fuel and pressurize it and send it to the injector? Also would there be air inside the fuel lines that way, which could hurt the system? Are there any examples of engines that dont utilize a lift pump and only use a mechanical plunger injection pump?
Virtually all diesel engines require a lift pump to get the fuel from the tank to the high pressure pump and into the engine. There is a noted oddity in recent history, that being the Duramax engine from GM. From its introduction in 2001 through 2016, it didn't have a lift pump (this is news to me too!). What makes this strange is, the injection pump on the Duramax is the CP3, which is virtually the same pump found in the Cummings diesel engines, yet Chrysler chose to use a lift pump with it. These are common rail systems, so there isn't a mechanical injector involved.
I would assume since the CP3 pump can function on a Duramax without a lift pump just fine, others could also do the same thing. I wouldn't recommend it. In fact, it seems there are a lot of people who add a lift pump on their Duramax equipped vehicle to help prevent CP3 failure. While a CP3 can last upwards of 200k miles without issues without one, it is my understanding they can last a whole lot longer with one.
As far as a mechanical plunger, most that I have seen pressurize the diesel at the head using a cam lobe to do the job (the Detroit Diesel 71 series, for example). I do not believe these have the ability to suck fuel, but need some sort of fuel flow to them in order to work correctly and efficiently.
As far as air in the lines, a lift pump will not affect whether there is air in the lines or not. Air is usually entered into the lines if the fuel level gets too low or there is some sort of breach in the system where air can enter. Air in an injection pump is bad news. Remember, diesel isn't technically a fuel, it's an oil. It's a byproduct of the distillation process to create other products from crude oil. Since its an oil, it also functions to lubricate internal parts of the injection system. If there's air in there where there should be diesel, running it for too long can and will cause damage.