This seems a broad question to me as the answer to both questions is **it depends** and answers can expand endlessly. As the community hasn't voted it as "too broad" I will attempt to answer although I'm afraid that it will be a "stapling" of other questions of this site.
##ATC 101
Let's start by clarifying one common misconception. There is not "one ATC unit" there are several: When a plane is close to a controlled airport it's controlled by tower. After takeoff by approach/departure during cruise by en-route control ("Centers" in the US, "ACCs" in Europe and I don't know for the rest of the world) and during landing it's again approach/departure and then tower again. For more information you can [have a look at this answer][1] which explains these concepts in depth.
This is irrelevant to the question but why do I mention that? **Because an approach/departure controller's job is significantly different than the one in ACC.** As a result they assign headings and altitudes in a different manner.
##Altitude
###Valid altitudes
[abelenky in their answer][2] mentions the "even-odd rule" for altitude assignment which is correct. But if want to go deeper this is not entirely true: above FL410 ("41000" feet) every flight level every 2000 feet. So 410 is the "odd" and 430 is the "even" (ie the westbound). And that was after [RVSM][3]. Before RVSM flight levels were every 2000 feet from FL290 up to FL410 and every 4000 feet above FL410. Again the "even and odd" was redefined. More information on RVSM on [this answer][4].
There are cases where the pilot can request to be exempted from RVSM. [Severe turbulence is one case][5] although I've heard that disengaging the autopilot is not the first option, it's altitude change request I guess mostly because ATC can easily [decline you the request to be exempted][6].
Also air traffic controllers can offer to pilots an opposite<sup>(1)</sup> flight level (specifically mentioning `opposite` in the phraseology) where the pilot can either accept or refuse.
###Altitude assignment
Usually when a flight receives a departure clearance, they are assigned a runway, a SID and an *altitude*. This altitude might be dictated by many factors like the SID itself, the terrain around the airfield and the traffic around the airfield at the time of takeoff. For the last case, the controller's judgement call is required.
After departure the controller will try to clear the flight at the predefined [handover][7] altitude with the ACC. If that cannot be reached on time a coordination needs be done between the 2 controllers.
###Climbing
When the aircraft is in the ACCs control, the controller will again try to direct the aircraft to it's final requested level. Usually upon first contact of a departing flight the controller will ask `foobar flight say final requested level` and the pilot will report it back. The requested flight level (RFL) is included in the flight plan, but for some reason that I've never asked they ask the pilot when they get them on frequency.
###Descending
After that and until the [top of descent][8] there is no significant change in altitude. At the top of descent the pilot will request a descent and the controller will grant it whenever it's safe and possible. And then the reverse process starts: get the flight at the predefined altitude for handover with the approach and then the approach will try to descent the flight to the appropriate altitude for engaging the ILS or other instrument approach procedure.
##Heading
###Departure
Before departure a flight is assigned an SID and they are supposed to follow it, unless they receive an instruction from the controller to deviate. When discussing with air traffic controllers in LGTS, they told me that the majority of the flights are assigned a "direct" to the TMA boundary. They assign a heading only if there is traffic in the vicinity and want to steer the departing flight away of it.
###Arrival
During arrival as it is correctly mentioned by [fooot's answer][9] a flight is (usually) assigned a STAR which they follow in order to engage the IAP that will get them safely to the runway. In airports with RADAR service though the approach controllers are vectoring the flight in order to create a stream of aircraft, a queue for landing. In order to keep this post short (too late I'm afraid) and because I don't want to repeat myself, I invite you to read my answer on the topic [here][10]. Then you can see if you still have blurred areas and you can post a new and more specific question.
###En-route
During en-route control the flights [fly on airways][11]. The routes they follow though [are not always great circle arcs][12]. So very often the pilots are tempted to request a direct to point to save some miles, and along with that fuel, emissions and time (so everybody is happy). The controller again might or might not grant the direct depending on the situation.
##Separation
All the above make the profession of the air traffic controller sound very easy: just a cool and relaxed guy giving predefined altitudes and headings. Nice... But that's far from the reality. Many times things don't go as planed and flights on the same altitude converge and the controller has to take actions. They have to evaluate each case and decide which technique they will follow to resolve the conflict: *altitude* change or *vectoring*?<sup>(2)</sup>
Even worse there might be 2 flights *not* on the same altitude climbing or descending and still converging and these are more difficult to detect. Remember when I mentioned the differences between approach and en-route controllers? Here is a good opportunity to explain it: An approach/departure controller (usually) has a smaller area to control than the ACC controller. Both vertically and laterally. That's why they can grant a climb from 4000 or 5000 feet to the upper limit on the TMA (which might be up to FL250) more easily. But an ACC controller has to be more careful: once they receive the flight they have to climb it incrementally. At higher altitudes the climb rate degrades and the speed increases rapidly. The plane will take *longer* to reach the RFL. If they grant the RFL very soon, how can they be sure, that many nautical miles away (might be from 100 to even 200 depending on the circumstances) there isn't an opposite or crossing flight that will come by and bite them?
Believe it or not this is the short version of the answer. You can take each and one of the paragraphs and expand them to infinity. Feel free to post new and *specific* questions if you would like to know any details. As you can also see from the links, most of the questions are already asked, so feel free to make a search first. I hope this is helpful.
<sup>(1)</sup> westbound FL for an eastbound flight and vice versa
<sup>(2)</sup> and some times speed control but let's keep it short(er).
[1]: https://aviation.stackexchange.com/a/25212/878
[2]: https://aviation.stackexchange.com/a/53631/878
[3]: https://www.skybrary.aero/index.php/Reduced_Vertical_Separation_Minima_(RVSM)
[4]: https://aviation.stackexchange.com/a/36113/878
[5]: https://aviation.stackexchange.com/a/5132/878
[6]: https://aviation.stackexchange.com/a/162/878
[7]: https://aviation.stackexchange.com/a/44913/878
[8]: https://aviation.stackexchange.com/q/1971/878
[9]: https://aviation.stackexchange.com/a/53633/878
[10]: https://aviation.stackexchange.com/a/45145/878
[11]: https://aviation.stackexchange.com/q/11506/878
[12]: https://aviation.stackexchange.com/q/13428/878