Level 50 questions

[Niwashi]

1) I notice I no longer get any xp for killing stuff, is that it, can I go no higher?

As mentioned, to raise the level cap from 50 to 60 requires the Heavensward expansion. (But there's a ton of content at level 50, so it's up to you if you want to do that right away, or finish more of the 50 content first.)

2) when I hit lvl 50 the game told me I could now accept the next class/job quest, cool but if I do them can I go up a class or not.

I'm not quite sure what you mean by "going up a class". There is another job quest at 50, just as there's been each five levels on the way up. From the level 50 one you gain the final piece of your artifact armour and a new skill.

3) Now I am at lvl 50 with my dragoon should I push on and finish the story missions or level up another class? Bit of a daft question I know, but I am really enjoying the story and since this has been my focus since a came back to the game I am a little apprehensive to finish it now, as then what will I do?

I'd recommend continuing with your MSQ at least through "The Ultimate Weapon" quest (the one that includes the Praetorium dungeon) which completes the original 2.0 portion of the main scenario, because doing so unlocks all the rest of the level 50 content. Whether you then want to branch out at that point either to leveling other classes or doing some of the optional level 50 content, or continue further in the MSQ is up to you (since you'll likely want to do each at some point). The main scenario itself continues on through more level 50 quests that were added in the 2.1 through 2.55 patches, (and then beyond that it continues with the Heavensward expansion).