"Turn an Old Router Into a Wi-Fi Repeater" is not a great solution because as mentioned the speed of the repeater wifi connection will be much slower than the main wifi connection.
If you can hard-wire the second router to the first router, a better solution would be to use a second router and set the SSID to the same name as the main router's SSID and set the channel to a different channel than the main router's wifi channel. Your devices should automatically switch to whichever wifi connection has the strongest signal.
I set my house up like this:
My main router, Asus RT-N16, is running Tomato because the QoS, which I use extensively, is infinitely better than DD-WRTs QoS. I have a second, older router, Buffalo WHR-HP-G54, that's hard-wired to the first and on the opposite side of my house, running DD-WRT using the same SSID as the first, but a different channel. Works great!
An added bonus of using DD-WRT on my second router is the ability to set up a second guest wifi connection that uses a different subnet so when guests are connected they can only use the internet and have no access to the network.
I can vouch for the TP-Link WPA4220KIT.
My parents live in a long apartment and their router can only be at one end. We needed wireless coverage on the other side of the apartment as well and we used this powerline adapter to boost our Wi-Fi.
Now this particular adapter is interesting because one end takes in an Ethernet cable and the other end clones the Wi-Fi network and broadcasts it as a router, so you don't have to resort to Ethernet or a get a second router for the other side.
It was affordable and very very painless to set up too.
Any tricks to enhance my computer reception?
I live on a building and we have a shared wi-fi and I would like to enhance my Mac's reception so it wouldn't be dropping all the time...
You may possibly be able to use the concept of the reflector from tip #4, but it may not work. You may need to buy an external wireless adapter with a directional antenna or perhaps use tip #4 on the adapter.