For those who experience problem accessing internet after certain time using TM Unifi router.
Before you reboot the router to "fix" the problem, try spending a moment to troubleshoot by changing your device to manual DNS setting, e.g. to google public DNS 8.8.8.8

See if you are able to access internet after DNS setting change on your device. In my case, it works if I don't query the router for DNS request.

I noticed this problem with TM Unifi router if your device is configured to use the router as DNS server. e.g. 192.168.0.1 (which is the default DHCP setting). The router seems to have problem serving clients' DNS request after certain period of time.

Best is to disable DHCP on the router, and setup a different DHCP server on your network. I have a second Wifi router in my network (for WIFI roaming in my home) and that serves as the DHCP server to assign google public DNS server to the clients.

Without serving as DNS server, my internet uptime with the TM Unifi router is superb, without speed slowdown and etc even after a long period of time.

A DHCP server can be a network router, or a PC/laptop. Network router is the best since it is mostly turned on 24x7.
If that is not an option, you can setup any PC or laptop to be a DHCP server (the one which mostly stays up & running on your network).

Here is the How-To for Windows machine.

If that is not an option as well, then no choice but you have to spend some money to get a better Unifi-supported router already.

Sure

Yes, confirm, IF it has been flashed to OpenWRT firmware. It works for Hypptv too.

Here is how to:
http://myjowenunifi.blogspot.com/2013/01/

Yes. Use the TM router to dial the Unifi (connect to BTU), then connect your second router to TM router (either via LAN cable or Powerline) as extension.

My setup at home is like this as well: BTU + TM router at upstairs, & 2nd router at downstairs, whole house is wifi covered. All devices able to do wifi roaming when at different floor without problem.

BTU = broadband termination unit

I am using Innatech RG4332 router, it is under WAN setup page. Please note that by default it is only configurable using operator account, not the admin account.

operator : h566UniFi