If I have target locked(action bar) a ship and then it moves away from range 3 this turn. The following turn a new ship comes within range 3 (360 degree) of me can I target lock (action bar) this new ship and remove the old target lock from the previous targeted ship?
You can always acquire a target lock on a new ship for your action, even if the previous ship was still in range.
Yes, you can do that.
Also, if you Target Lock a ship and it moves out of range 3, you don't lose the that TL.
If I have target locked(action bar) a ship and then it moves away from range 3 this turn. The following turn a new ship comes within range 3 (360 degree) of me can I target lock (action bar) this new ship and remove the old target lock from the previous targeted ship?
Certainly.
The Target Lock doesn't even need to come from the action bar although many of the other ways of acquiring a target lock have some kind of restrictions.
I will admit that I am not 100% certain what happens if you acquire a new Target Lock when you already have two via Weapons Engineer. Now that upgrade may allow you two new locks when acquiring a TL but if you only lock one ship I believe you can choose which current TL you will let go of for that new lock.
Weapons Engineer is a "may" ability, you don't have to use it, so you can certainly only acquire 1 lock.
I'm not sure if the rules actually specify a procedure to follow if you somehow acquire more TLs than you can hold, but the logical way to do it would be "If you ever have more TLs than you can hold, immediately discard TL pairs until you have the allowed number" or language to that effect. So, if you acquire a lock while you already have 2, you then have to discard one of them of your choice.
I don't have the rules in front of me, but I think it's the other way around: you must discard old TLs first to make room for the new ones. Not that there are any gameplay implications I can think of that it would make a difference.
I don't have the rules in front of me, but I think it's the other way around: you must discard old TLs first to make room for the new ones. Not that there are any gameplay implications I can think of that it would make a difference.
The core rulebook, page 9 implies that they happen simultaneously: