I don’t know the term for “the thing that holds the router”. But that contraption is essentially attached to the Z-axis “thingy”. You know - where you attach the router. Anyway - I can pull mine forward away from the gantry from the bottom of the router a little. About 0.15". It appears as though when carving, the entire spindle ‘tips’ when the bit contacts the wood.
My question - is it normal to have that much play? Should I be able to tip that thing forward at all? If not - what should I tighten?
Regardless of which machine or Z you have that is not normal. Your machine must be tightened up and trimmed or you will never be happy. You are going to need to take a photo of it and someone will identify what you need to do. Instructions will vary depending on your setup. A serial number will help, too.
To add to your info,. I check my machine every 3 weeks. Every bolt, every wire connection. Never found an issue, that also means I shouldn’t expect to see one during production.