Since there is only one copy of each event, am I understanding right that it is played upon purchase?
Yes.
Not quite. They are never played. The Event card stays where it is. The Event (which is the thing being bought) does what it says when you buy it.
With reserve cards is it discard pile upon purchase then once shuffled in play the top half such as (+1 action, +1 card) or whatever. Then during clean up placed on reserve mat to be called on to use at beginning of turn when needed?
Mostly right. On play though, reserve cards technically go to the Tavern mat immediately, even if in practice you may want to wait until clean-up for this. This is relevant for cards such as Duplicate and Royal Carriage, which can be called at times other than the start of your turn. You may play a Royal Carriage, move it to the Tavern, play another action, and then call your Royal Carriage immediately to replay it.
It also matters for cards that care about what you have in play, like Peddler, Mint, Bank, Horn of Plenty, Mandarin, Bonfire, Pilgrimage, Royal Carriage...
I would suggest not keeping it in play until Clean-up, but rather putting it close to your Tavern mat.