As someone working in the mac repair field, I think I can shed some light. Only Apple Authorised Service Providers have original Apple batteries, and they charge a premium for it. A Macbook Pro 13" 2009-2012 (non-Retina) model will set you back for around $400. It should be around $140 - $180 with third parties.
As for the third party repairers, there are no original batteries out there, despite some who claim they are originals. These batteries are what we call refurbished batteries. The battery reuses the original microchip controller inside the battery, but the battery cells are brand new, and are installed in a brand new case.
The other kind is totally brand new. Uses third party microchip, new cells and new case. The appearance of these batteries are identical to the originals. Performance wise, both are identical to the originals. I would rather get the non-refurbished batteries actually. These are what iFixit sells. The issue with the so-called originals which uses genuine Apple micro controller is that these chips are old, and sometimes will malfunction and report that your battery needs to be replaced when the cells themselves are still fresh.
The third kind, which are the cheapest of the cheap, are batteries that differ in appearance from the originals. They are those sold by the 2 largest online battery sellers. Totally no brand. How you can tell is the price. They are usually under $100. These batteries uses low quality cells and are very light.
Even original Apple batteries bloat up like mad. See this
ONE
Everything is made in China these days, and no, batteries usually don't explode, so it is safe to buy China made ones. Bloating happens because the oxygen mixes with the lithium inside, which usually does not happen, but sometimes due to wear and tear, the metal casing enclosing the lithium gets punctured.
Magsafe chargers wise, yes. Chargers might spoil and end up in smoke. I don't recommend anyone to get third-party chargers.
One thing to look for is the warranty. Repair shops confident of their batteries will offer 1 year warranty for their batteries.
Hope that helps!