Fladude:

In the price range of the DRs you mentioned, I'd recommend the Heym first, followed by Chapuis and Merkel. If you can get comfortable with the safety system, and they way they handle, the Krieghoff is a decent quality gun. In terms of overall quality, I think the last three offer quality commensurate with price, while the Heym offers better build quality than the price suggests. For the build quality offered, I view the Searcy as overpriced and don't recommend it.

Regarding your question about the actions types, the boxlock is a "body action" gun with it's lockwork contained within the action body, or "box". The lockwork of the sidelock is attached to the lockplates behind the bar and detonating. There are two types of sidelock, bar action and back action, so called because of the location of the mainspring. The mainsprings of the bar action lie in the bar of the action, in front of the lockwork, while those of the back action are attached to the lockplates, and lie behind the lockwork. The back action has long been generally preferred for double rifles because the location of the mainsprings allow leaving more metal in the action bar under the detonating, making it theoretically stronger for the higher pressure double rifle application. However, plenty of best quality sidelock double rifles have been built on bar actions, Purdey's and the British Rigby's come immediately to mind. Hard service "working" DRs tend to be boxlocks, while so called "best" guns tend to be sidelocks, but there are exceptions to both.

Don't attempt the sidelock route on a budget. The good ones have never been "affordable". A top quality boxlock is preferable to a cheap sidelock that's available at the same price.