Steve, the first thing you need to do is download (free) Honda's short-form workshop manual for your engine, from here:
http://www.honda-engines-eu.com/en/images/1007.pdfYou will see those rubber tubes on page 66: they both come from the carburetor area. It looks to me as if one of them is for an air vent to the float bowl, and the other may be for oil drips from the breather system (something that tends to happen when the engine has been overfilled with oil, or has developed blow-by). The tubes are just to ensure that dirt does not get into the engine through the vents, eventually causing a need for maintenance.
So far as the speed variation is concerned, where a governed engine (like yours, and nearly all mower engines since the 1960s) has moderately lean mixture, it often tends to hunt: that is, cycle in speed, with a cyclic period of maybe 5 seconds. The reason is that the carburetor has an acceleration/power enrichment function. Running lean, the engine slowly loses speed and the governor opens the throttle slightly but this has no effect because of the lean mixture. As speed continues to drop it gets so low that the governor opens the throttle wide, and this causes the mixture to become richer temporarily. The engine responds vigorously with a speed increase, the governor closes the throttle, and the whole process starts again. The underlying fault is that the carburetor is running lean at light throttle (very light, in your case).
Your engine is designed to idle at 1,400 rpm. If you have tried to make it idle slower, hunting is one of the possible results of this. Also, many Honda engines have a small adjustable air bleed in the carburetor (called the pilot screw), which is capable of making them a bit richer, especially near idle. However yours being an emissions engine, this screw has a sealing cover over it, and is designed so that tampering will damage the screw. It looks as if your engine needs a tune-up, as described in the manual. If all else fails the carburetor may have to be cleaned internally, but I suggest you go through the Honda maintenance procedure, which is non-intrusive, and see how that works out before taking anything critical to pieces.