I understand that you can use your starting merit dots to buy an item like an artifact or imbued item in Mage the Awakening, but what about afterwards? Would I have to pay experiences if I were to find one on a defeated foe or craft myself? Could I chose to spend experiences to buy one I designed between sessions or during downtime?
Would I have to pay experiences if I were to find one on a defeated foe or craft myself?
While I do not have all of the available supplements, I am not aware of any rules directly addressing this. Thus, it is in "ask your storyteller" territory.
With that said, as a storyteller, my interpretation of the rules and my ruling has always been that a player does not have to pay experience for any merit, including artifacts or imbued items if they acquire it during play.
However, items acquired that way are not protected by the sanctity of merits rule (See page 99 of Mage: The Awakening 2e). Also, while I believe that my interpretation is the intended one, the language of the Sanctity of Merits rule could be read to imply that a player should pay experience even for items acquired in story, at least if they want those items to be expected to stay around long term and have a meaningful mechanics effect.
While I do not do it, a Storyteller would be well within the cannon to have either the Mysterium or a higher up in the character's own order demand that any artifacts be turned over if the character didn't pay the experience cost for them once their purpose in the campaign was served. In fact, for certain artifacts received under certain conditions, it would not be at all surprising for it to simply vanish after its purpose was served, recalled to the supernal realm itself after serving the purpose it was sent to the fallen world to achieve.
Could I chose to spend experiences to buy one I designed between sessions or during downtime?
With storyteller approval, yes. Merits cost 1 experience point per dot. (Page 83 of M:tA 2e). With that said, the storyteller has the right to outright veto or add in-story requirements to any developments in game. This would especially apply to a custom built artifact or imbued item. At a minimum, you would expect even a very lenient storyteller to determine the item's rank and thus cost rather than expecting the player to decide that.