The short answer to most of your questions is: do not set IPs statically. You haven't actually given any real reason why you want to do this anyway, so all the more reason not to do it. But basically, the system uses DHCP for several reasons, one of which is pnp as you say - you do not want to try to set devices up manually, esp if you have holes in your LMCE knowledge (and networking knowledge, which you mentioned).
One significant reason is, if you have problems (which you probably will as you are re-engineering the system) you will not get much sympathy on the forums if your first step was to change the design
There are almost no meaningful reasons not to use DHCP. The only one that some people get hung up on are you may have fixed firewall rules port forwarding to specific IP addresses or the like - but this doesn't really hold up, because once LMCE has assigned an IP address, under normal circumstances it will never change 1) because the client will continually request a renewal of the same address and 2) because LMCE records the MAC address of the client's NIC and always had out the same address anyway. So they effectively become fixed addresses anyway.
1) You don't have to throw out the router if you don't want to. That's up to you - but it is true you won't actually need it anymore (are you sure the ADSL modem isn't a router anyway, most are, in which case you never needed it anyway). Yes, under normal circumstances your entire existing network would be behind the LMCE core/hybrid on the "internal" network.
3) I have never come across a device (in many years, at least) that does not support DHCP.
4) The core/hybrid acts just like any other router except that it has a firewall on it so for certain things you may need to set up firewall rules. In short, a router doesn't set up "IP maps" it has "routes" that describe entire subnets (eg 192.168.80.0/24) is reached through interface X, via gateway Y. There will always be a "default route" or "route of last resort" that tells it where to send traffic that doesn't match any of the other, more specific routes. Essentially, your core will have 3 main routes, one that says 192.168.80.0/24 subnet is out the "internal" NIC, one that says your external subnet (something like 192.168.1.0/24) is out the external NIC, and one that says all other subnets are out the external NIC via the internal IP address of your modem/router.
5) Definitely set up with Internet access, it will make several things easier/possible. After you have installed, you don't need to maintain that Internet access if you don't want to.