TV and Internet

Not all receivers work with all satellite dishes. For example, Tailgater (for Dish Network) uses a Dish 411 or 211 receiver.

We have a Dish 211v receiver hooked to a small TV in the kitchen. We take that receiver with us when traveling in the fifth-wheel.

Yes for TV but not for internet. Satellite TV will come from DirecTV or Dish but internet will need to come via some cellular provider. This will NOT be anything close to what you now have assuming your internet is provided by your cable provider. That doesn't mean it will be bad but it will be slower and it will be location dependent. It will also likely be more expensive for less with data caps that you don't have to deal with from a cable provider.
You can setup either satellite service at both houses with antennas permanently mounted on the (assuming you have a good clear view of the Southern sky) and then you will need another antenna for the RV, either self pointing or manual and either roof mounted or portable. What services you want will make a difference in which antenna is required to meet your needs. You will then be able to move the receiver from home 1 to RV to Home 2 and back again as necessary.

The only thing that might not work with sat TV is local channels. That can be fixed with a simple phone call.
As for internet, the easiest is cell based wifi. Not the cheapest, but definitely the easiest to have available all the time. Verizon followed by ATT offer the best overall coverage

Our motorhome is our home, but we also have a small cottage in upstate NY near our kids that we visit from time to time. We have Dish satellite TV service with two Hopper w/Sling multi-tuner DVR receivers, one we keep in the motorhome and one we leave in the cottage, both on the same account. The cottage has its own permanent dish, and we carry a portable dish in the motorhome. With the two Sling equipped receivers, we can record programming at the cottage that we would miss while underway, and watch it later in the motorhome using the Dish "Dish Anywhere" app. When we visit the cottage, we just hook up to the dish there, and both Hoppers can see everything recorded on either one. The only downside is that when we change our service address while on the road to get the current locals, the cottage Hopper looses its satellite locals. The cottage does get the locals over the air though, so it's not really a problem.

Yes, just carry your home receiver (or buy a separate receiver if you wish) to you RV. It will cost you about $7.00 as mentioned above.

I used my home receiver in my m/h for years. I now pay $7 a month to have a separate receiver in the m/h.

Winegard makes an auto align carryout dish for DISH that gets HD for about $400.

Yes... that's what we have. Dish Satellite service at home and use it in the RV. Works great.