It's a good question, @NewFlyer78 .
What you should remember at the outset is that the whole OHJ was designed in the day when few aircraft had radios and all airfields had signal-squares. Therefore the standard way of approaching an airfield was to fly over it at more than 2000' agl (in other words, outside of its ATZ vertically-speaking) so that you could look down upon it from above and determine runway in use, circuit direction etc.
Once you'd done that, you would then position yourself to descend into the ATZ (ie, enter the ATZ from above) but on the deadside (ie the safer side) to then join the circuit at circuit height (ie, the same height as everyone else) by passing over the upwind numbers on the runway in order to join cross-wind, which was considered to be the safest place; bearing in mind that few aircraft taking off would have the climb-performance to be at circuit height by the time that they passed the far end of the runway. That was the theory and, despite its many nay-sayers, it works just fine.
So, to your specific question: your first call should be:
"Tumbleweed Radio, G-ABCD, approaching from the [west] to join overhead." They may or may not acknowledge; they might well give you the runway in use and QFE. They might not answer at all.
When you are in the overhead (and it doesn't really matter precisely where you are so long as you are over the top of the ATZ) you say "G-ABCD, overhead, descending deadside". Then everyone knows where you are and what you're doing. Again, you may not get an acknowledgement.
Once you descend into the ATZ you are obliged to make all turns in the circuit direction. This, whilst sensible, presents all sorts of challenges to students as they struggle to position themselves such that they descend on the deadside in an appropriate wide turn. It just takes a bit of thought and practice walking around your living room with a bit of paper on the floor to represent the runway and the circuit direction. You can hold your arms out like an aeroplane if you want
Once you have joined the circuit and have turned downwind you call "G-ABCD, downwind to land".
As with all things, keep a very sharp look-out. Someone else may be joining overhead also.
To your second question about departing. Again, go back to first principles. All turns in the ATZ should be in the direction of the circuit. So, if the circuit is a left-hand circuit but you want to turn right then you have three alternatives:
1. follow the circuit round in order to either depart off the base-leg at circuit height or, as a variation, continue climbing in the circuit and, once above circuit traffic and only if safe, turn to depart from the overhead in the climb - both are acceptable.
2. climb out straight ahead until you are outside the ATZ (ie more than 2nm from the centre of the airfield) and then you can do what you like (assuming that you aren't in anyone else's airspace, of course).
3. state your intention to make a non-standard departure, in this case to the right. You aren't asking permission (because an a/g can't give that permission) but you are informing them and everyone else of what you intend to do. Some might argue that such behaviour would be slightly discourteous - and so it is probably the least favoured departure option. If you were considering it it would do no harm to ask at the desk as you book out and you can then ascertain how they would view such a departure. Good manners go a long way in aviation.
Finally, as has been mentioned, many airfields will have local noise-abatement procedures which you are expected to follow. After all, you are a visitor but they live with their neighbours there 7-days a week. Quite often these procedures may prohibit a non-standard departure. So check before you take-off.
Hope that helps.