Cooking: We used a Jetboil Helios for Another beautiful Lap. It was more than enough for our needs. This meant we did not have to carry a heavy Gas bottle and have the grief of having to store it outside the vehicle. We also have a couple of Jetfoil Flashes.
I also have an MSR Whisperlite from my old bushwalking days. A very reliable and great little stove. It still gets used now and then.
Tents: For ABL we started off with an REI Kingdom 4. A great tent and very roomy. Later in the trip we purchased an ARB rooftop tent. This was mainly to get up out of the dirt and sand of central Australia. A great tent although the mattress is a little firm and the fly does flap a lot in the wind.
For bike trips I use a MSR Hubba or a North Face Moutain 25 depending on the season and location. The Hubba is great but its very small with nothing in the way of a vestibule. The Mountain 25 is a 4 season mountain tent with lots of room and a good vestibule but its a fair bit bigger to carry.
Sleeping: Jaks has a North Face Blue Kazoo and is very happy with it. I have been using my old Macpac Neve for a long time. It has no down in the bottom so packs up very small. Time to update and I think it will be to an Exped Waterbloc 800. When I am bike camping and using the Hubba it does suffer from condensation a lot. Its very hard to get the fly off without wetting everything inside. If I can get it without bumping the fly its all good but that does not happen very often. The Exped is waterproof so it won't matter if a few drops land on it and it has the bonus of being able to be used without a tent at all.
I have an Exped down sleeping mat. I have used Thermarest ones in the past but the Exped is much warmer and more comyfy. For car camping we used a couple of thick Kathmandu mats. Bulky but very comfy.