Mage Hand has a duration of "Concentration" that is, it keeps up as long as you're devoting a standard action to it each round. If you spend a burn on telekinetic haul, for the next [level] minutes you won't need to spend that standard action, and the weight you can carry increases for [level] minutes.
"Unattended Object" is one of those things that you need to ask your GM about. Expect table variation.
Dunno about the TK, but beetles are objects not creatures (despite being alive) so they cannot attend anything. If there's a rabbit or a PC (or an intelligent beetle) on the rock, OTOH, then
An item attended by a character (being grasped, touched, or worn) makes saving throws as the character (that is, using the character's saving throw bonus).
is AFAIK the only "official" definition of attended (so the rock is being attended by whoever's sitting on it).
Hope that helps.
Hrm, I can find places that specify that ordinary plants lack Wisdom & Charisma scores and are therefore objects not creatures, but I can't find anything saying that for ordinary insects. Anyone know whether random beetles are objects or vermin?
You could just cast TK haul on the item before it is attended. The spell never says the item has to stay unattended. That language is basically just there to keep you from stealing stuff out of people's inventories.
"Touch" in pathfinder (and in English) is not restricted to direct contact with flesh. A blanket will not stop you from touching something any more than gloves will, and you can definitely apply touch spells through gloves. Otherwise you could just have the party stand on the whatever in their boots and socks, or sit on it in their pants.
Having everyone (including you) get on the carriage/boulder after you start TKing it should work fine, though.
I really would like a blog post on objects. I have questions like, "How small can an object be and still be considered an object?"