Usually the case, no doubt...I've been tempted by these cheap clusters (ooooh digital and LED backlighting:welcomingbefore and this adds further doubt as to the quality of them. I'd love a digital gas gauge too, IF it were reliable and yes, a gear indicator would be nice too.
Are you sure it was wired in correctly and not going over (or under) voltage? If you get a new one I'd strongly suggest waiting until you get your zeners in to re-connect it. Posplayr knows best. It sounds like it may have cooked.
I buy a lot of inexpensive items from China off of ebay to open them up and see how they are put together. Generally the stuff is usable if you really have a use for it. The main problems I see are ultra thin wires, which in reality are probably properly sized electrically but just don't have the mechanical strength of larger wires. I also open them up to inspect the workmanship and see if there is anything "just hanging on". Most stuff seems to have a QC stamp and was tested or inspected going out the door (but no guarantees as to what it really means).
If it was working and then not working, check for a fuse and any other power connections. You have to be careful when wiring this stuff up so that you don't inadvertently short out wires. It might be that a power wire broke off from the internal PC board.
Email the vendor, although they will probably want you to ship it back to China for a refund.
About the Zener, the RPM input should have some protection to start with and why it needs more I don't know. So my point is while the zener should make the RPM more accurate, I would doubt that the device would be damaged if you did not have it installed.