First, you must confirm where's your fibre distribution point. If you've confirmed than it should be fine.
Have you forgotten your phone line? Connected to the BTU. If you wanted to place it in the living room, you might need to conceal cable for it too. Other than that, I don't see serious problem but you must plan how your wall socket(RJ45 port) will be installed. Since from the first room, you wanted to connect to 2 more rooms and to living room. That means you'll need 3 ports + 1 for telephone at the first room and 1 port each at remaining rooms and living room. I have no idea how are you going to lay the cable. The most hassle free solution I would say is to use wireless. Concealing 4 cables ain't so easy IMO. Anyway, just my 2 cents. With current wireless technology, everything is quite stable. I'm 99.99% on wifi. Nearly never use LAN cable to connect my PC and router.

Whats your plan Whats your objective? Do you plan to set up a NAS? Where do you want to put your equipment at (FWS,BTU,RG,STB)? Phone? Based on what i undersand, you want your IPTV to be in your living room right? Are you planning to use a repeater/AP? U want to put it upstairs/downstairs? or just run cable?

It is not advisable to place your network stuff in living room due to aesthetic reason unless you can conceal it in a cabinet. You can place it in a study if you have one or in the bedroom that house the BTU. I gather you plan to place BTU at the fiber incoming location. You can place both BTU & RG there and run 2x Cat5e to living room and 1 ea to 2 other bedrooms. You need 2 for living in case you also need networkfor appliances other than STB. You can't share them with STB unless using VLAN trunking and that would cost more. You can run more Cat5e to other rooms such as downstair backroom and kitchen for future smart kitchen appliance. You can add switch to the whole setup when you need it in future. Myself have network points in 9 different locations with network centre located in family room. In case the RG location is not suitable due to inadequate wifi coverage, you can always move it to living room and place a switch at its original location to link it to rest of the room. Better still, buy a better RG than those supplied by TM.

Which phone are you referring to? The VoIP through UniFi or your Old Phone? You need to tell this in order for us to help.
Why don't you just do the installation at your living room and conceal just cat5e from living room? Btw, cable must confirm working and better get some higher quality ones. Else if the cable is broken after being concealed... Your heart sure pain like hell 
Anyway, yes, google dns/open dns can allows you to open file sharing sites.