"The router will forward a packet sent by host1 down its default gateway if and only if for all it's connected interfaces, the interface's configured address bitwise AND with host1's subnet mask does not equal host1's network address"?
In traditional IP routing, yes. Packet forwarding is done by choosing the most specific matching route from the routing table (there is no special distinction between "connected interface" vs "default gateway", they're both just standard routes). So if the router has two routes matching the packet, one for
0.0.0.0/0 (default gateway) and another for
22.214.171.124/24 (local subnet), the latter always wins.
Although you should note that routers don't necessarily have a default gateway (especially in the so-called "default-free zone", which really just means they have more-specific routes for absolutely everything).
Also, I say "traditional IP routing" because routers might do more than just destination-based route matching. For example, in IPv6, some systems support routes which match against source and destination; e.g. you can have routes
from ::/0 to ::/0 via <gw1> and
from 2001:db8::/48 to ::/0 via <gw2>.
It is also common (in both IPv4 and IPv6) to implement policy-based routing which might match packets by protocol, or by firewall mark. If a router has two upstreams (two default routes), it'll often want to use policy routing to ensure that packets received from upstream 1 will have their replies sent to the same upstream 1, even if the routing table would prefer upstream 2.
In particular, if the destination address is on the same subnet as host1, say 126.96.36.199, but not existing on the local network, is it still true that the router will never forward it down the default gateway?
Yes. If the routing table says the destination is local but the router fails to receive an ARP/NDP reply, it will not fall back to another, less-specific route – it will immediately give up and return some sort of ICMP error, probably "Destination host unreachable".
(Whether the destination is in the same network as the source usually doesn't affect anything. Even if it means sending the packet out the same interface as it arrived in, the router still behaves the same way.)
Finally, if I (absurdly) choose a subnet mask of /0 for host1, [...] the router would be acting as a switch and spamming every packet to all connected interfaces (except the gateawy interface), and never forwarding anything down its default gateway?
Changing the subnet mask on a host has no effect on the router. The router doesn't know what subnet mask is configured on other hosts; it'll behave according to how its own interfaces are configured. (And vice versa, the host doesn't know how the router is configured.)
If you directly configure a router to have a /0 subnet mask on some interface, it absolutely does not start acting like a switch. It still acts as a router, but if it had a default route before, it simply has two default routes now – one of them just happens to claim that all hosts are local on the specified interface.
If the router prefers this second default route, you'll still see the usual "local subnet" behavior: the router will keep trying to ARP/NDP every destination directly on this interface (assuming it's a broadcast interface like Ethernet/Wi-Fi), and still won't forward the packets until it has received an ARP reply and learned the destination's MAC address.
And if you configure a host to have a /0 subnet mask, basically the same thing happens: two default routes, one of them local, so the host will try to make ARP queries for every destination and fail. (The only difference between hosts and routers is that a host won't forward received packets while a router will. Otherwise they process routing tables in much the same way.)
Finally, if I (absurdly) choose a subnet mask of /0 for host1, this effectively blocks the whole internet from my host?
Normally yes, because the host will never succeed at ARP queries for destinations which aren't really local.
But you can have the router spoof replies to those queries – a feature called "Proxy ARP". With it enabled, when the host tries to make an ARP query for some distant host, it'll learn the local router's MAC address instead, and from that point everything appears to work as if the host had a normal gateway route. (The only difference is that the host will end up having a huge ARP cache.)
This is more commonly used when the host thinks it's on a standard subnet (say, a /24) but cannot actually send Ethernet frames to anything else except the local router. A hosting company might configure its switches to drop all packets except from/to "trusted" router port (port isolation), and enable proxy-ARP on the router. This allows traffic even between customers on the same subnet to be filtered according to the router's firewall rules, which they'd normally bypass.