1 - Are there other servers that maintain all domain names for a
single TLD (other than the root name servers)? Can I set my BIND
server up as one?
Any DNS server can host any zone it likes - you can even create your own root (".") zone if you like. However people will only query your server for this information if correct delegation is in place. This is done with NS records. It's common for a large company to create an internal root zone - but this should not be queryable by the outside world.
If you want to hold a copy of the root zone, e.g. if you're behind a firewall and can't directly query upstream nameservers, InterNIC provide a copy here. Note that this is just the root zone, which contains delegation for TLDs like com, net, org, uk; not the data within those zones themselves.
As far as nameservers for the TLDs go, these are all operated by different companies. Possibly not all of them provide a copy of the zones like InterNIC do with the root zone. Zones like com are massive.
2 - Is it possible to query root name servers for all recently created
top level domains?
Not using DNS, no. If you want to do that, download
root.zone from InterNIC regular and run
diff against it. You should also avoid querying the root servers directly, where possible.