In this case, "try" and "try out" both mean "to test/examine".
You have to look at what comes after the "try":
- try + noun = test: Shall we try the new restaurant?
- try out + noun = test: I want to try out the new Mercedes.
- try out + for = attempt: I'm going to try out for the football team.
- try + on = test clothing/accessories: Try on this perfume and see if you like it.
- try + infinitive = attempt: Let's try to get seats close to the stage.
Check your question to see how I edited your example questions.
All those expressions mean largely the same thing, with a few very subtle differences (connotations):
Please come over to my place tomorrow night.
This means "When you're someplace else tonight, standing or sitting, working or drinking, I would like you to start moving in the direction of my place; cover the distance that separates you from my place by car, by train, or on foot, I don't care. I'd like to see you. You're officially invited."
Please come by my place tomorrow.
This means "When you're on your way to wherever you need to be, by train, by car, or on foot, make a stop at my door, there's a good boy. Don't forget to ring the bell. I'll be home."
Please come up to my place tonight.
This means "When you're done with whatever you're doing today and/or tonight, just remember me and get to my place. It's north (or uphill) from wherever you expect to be at that point; you can't miss it. Bring a bottle of chardonnay, I'll make the sushi. Or would you prefer Italian? I like Italian. I think you like it too. Don't bring your wife or mistress, that would be uncouth. Come by yourself."
Come around my place tonight.
This means "Very casually, as if by accident, please end up standing in front of my door tonight. Don't forget to ring the bell. Bring a bottle of chardonnay only if you really mean it."
Something like that.
Best Answer
First and foremost, all four of those sentences are in fact grammatically correct. What Google demonstrates is that a.2 and b.1 are really awkward and therefore much less common.
In terms of the implications of the words themselves:
Opportunity often implies that it is an offer being extend or provided to you by a specific entity or person who is providing you with means or privilege otherwise unavailable to you. Sometimes there is no other party offering the opportunity, but there is always the sense of urgency, that it is a limited time deal, something that must be seized or it will be lost. There is generally a positive connotation: an opportunity is a good thing, something to be excited about.
Possibility simple implies that it is something that can happen, either because you decide to make it happen or because circumstances cause it to happen. There is no implicit positive or negative connotation here. There is also no implicit implication of the likelihood of the event happening. If it is possible, it may or may not happen.
In general, opportunities imply possibility, but not the other way around. If you are given the opportunity to go to Norway, then of course it is possible. But if you lived within driving distance of Norway, you wouldn't usually refer to your possibility of driving there at any moment as an "opportunity".
Some examples:
Interestingly, this Ngram suggests that in fact the usage of "opportunity to go" and "opportunity of going" have reversed in the last 200 years, with both being approximately equal in use around AD 1890-1900.