Need help connecting TV to 'man cave'

Do you have power already, in the shed?

Will it just be another cable TV 'drop' out there? I get basic cable without the cable box. Or do you want to use the cable box in the family/lounge room? Do you mind watching the same thing in your shed as is playing in the lounge room? Do you need a cable box in the shed?

You know what they say.... If your not in total control of the remote, your not going out to your man cave, your being sent out into the proverbial Dog House

Not that I or any friends of my associates would do this ... but if you want to save on the needed RG-6 direct burial cable :

Call your local cable company and tell them your building a house and have a long service trench from your house to the street, around 250 feet, that you must fill in tomorrow. Let them know you need to have them come out and lay their cable in the trench before that, so you can be a cable customer. They will inform you that there is no way they can get anyone out there with so little notice, but will typically inform you that if you could come by their facility, they would be happy to give you the needed cable if you would not mind placing it in the trench yourself. Not that I or anyone I associate with has ever done this, but, they will typically just give you a 500' spool given that needed distance.
You might want to have an address off hand to give them, as they may want to confirm your in their service area though. Or so I hear

You may want to pull some direct burial twisted pair also for phone / IR Repeater ..... this requires a similar call to your local phone company
Or so I hear again

Assuming your cable box is HD and has an HDMI output, my suggestion would be to use an HDMI over Cat5 extender. If you're already using the HDMI output from your cable box for the TV in the lounge, you can either use an HDMI splitter or use the component video output (which is also HD capable) from the cable box for the lounge TV (you'll also need to make a separate audio connection) which would free up the HDMI output to feed the other TV via the HDMI extender.

To control the cable box remotely, the Hot-Link-CAT-5-Repeater-System is highly regarded by many on this forum but I have no first hand experience with it myself since I use the Next Gen wireless remote extender which maxes out at 100' based on the specs.

Why don't you just steal it off one of their trucks?

Why not just run an RG11 from the demarc at the house to the man cave and add a new cable box? RG6 would probably work too, but 11 for sure. I would guess that the cable company was just saying that there is no drop to that building and they aren't interested in running one.

Exactly. The cable company doesn't need to know where the box is, or care for that matter. They don't install the boxes anyways, you do, so as long as it is at the same physical address, then you're fine.