London to Manchester at the end of BOE 14 days in Oct

1) It is too early to buy tickets to Manchester. They will typically go on sale about 12 weeks in advance. The direct train departs London Euston station and travels to Manchester Piccadilly station. Avanti West Coast is the train operator. When you are searching for trains in the UK, the National Rail website is the best one to start with https://www.nationalrail.co.uk/

You can take the tube from your hotel (Gloucester Road tube station) to Euston. Use the journey planner on this website https://tfl.gov.uk/

2) Since you are only in Manchester one night, I think it would be easier to choose a hotel near Manchester Piccadilly station. When you arrive, drop your luggage at your hotel. From Manchester Piccadilly you can then take a tram to Etihad. I typically use googlemaps to check transportation routes. I don’t have a specific recommendation for a hotel near Piccadilly station.

3) You will take train back to London Euston. From London Euston, you can take the tube/Elizabeth line to Heathrow. Use the TFL journey planner I linked above to see the route.

4). How early is your flight to the U.S.? Personally, I prefer to stay in London the night before my flight home so I can have a nice dinner and see a show. I don’t have any budget recommendations for T5 as I always stay in London. I’m sure others will have suggestions.

There is a non London route to Heathrow (well two actually) from Manchester to Heathrow which seems to be little known.

Train at 15 minutes past each hour to Watford Junction, change at Milton Keynes Central, then the rail air bus from Watford Junction to Heathrow. Admittedly that only serves Terminals 2 and 3.
The other even easier (but longer route) is the direct train from Manchester Pic to Reading at 25 minutes past each hour then the rail air bus. That bus serves all terminals.

There is one hotel attadhed to Terminal 5. It is the Sofitel but it is not budget.

There are a few hotels attached to Terminal 4--Premier Inn, Holiday Inn Express, and a couple more. These two would be consdered budget. You would need to take the tube from Terminal 4 to terminal 5 but its free. There are also numerous hotels just outside Heathrow which would require taking the Hoppa bus or a taxi to Terminal 5.

From South Kensinton to Eustons, you can either take the District or Circle Line to Victoria Station and then transfer to the Victoria Line to Euston. Or take the Piccadilly Line to Green Park and swtich to the Victoria line to Euston. The first way is probably faster.

In Macnhester, it's better to stay closer to Manchester Piccadilly train station. Better choice of hotels and restaurants.

What days of the week will you be travelling to and from Manchester? Do you have actual dates? There is obviously a chance that there could be a game on that night, unlikely but possible. But I assume that you are just planning a tour of the stadium.

You can clearly have quite a bit of time in Manchester if you leave early and come back on a later train. There are more than enough tourist sites to fill the time you have available (and for several more days). As football fans the National Football Museum would be an obvious one, but off the top of my head here are some more places to look up: -
John Rylands Library
Chetham's School of Music and the UK's oldest library
Manchester Art Gallery
Whitworth Art Gallery
Museum of Science & Industry
Lowry Centre
Imperial War Museum North

There are many more.

With regards to sights:

I took a walking tour with this company. I thought the guide was good and it was a good overview of the city. There was some coverage of the city’s two football teams during the walk, as well as history, music, etc. https://www.simanchester.com/

I also visited the John Rylands Library, the Central Library, the Art Museum, and the Science and Industry Museum.

I went to two concerts at Bridgewater Hall (one classical, one a popular singer). This is a beautiful concert venue. If you like music, it is worth checking to see what is on the night you are there https://www.bridgewater-hall.co.uk/

I stayed at Motel One, 34 London Rd, Manchester M1 2PF, which is a 4 – 5-minute walk from the Manchester Piccadilly Train Station and 12 minutes from the stadium if you take the Metrolink Red Line. I enjoyed my stay, and it was convenient to store luggage while I explored the city.

As Laura mentioned, being near the train station will make it more convenient dealing with luggage, etc.

There are several restaurants for dinner and breakfast near the train station and in the station. You can walk to several interesting neighborhoods and sites from the Motel One and use the tram as well to venture farther out.

All the best,

Forget looking to stay near the Etihad, book a hotel in the city centre, preferably near Piccadilly station, but in reality any hotel in the city centre would only be a maximum of 20 minutes walk to Piccadilly anyway.
Take the tram, known as Metrolink to the Etihad. The tram stop is beneath Piccadilly station, take a tram with a destination of Etihad campus or Ashton. It's only three stops and drops you off right next to the stadium. Forget about the colours of lines, locals don't use them, preferring to use the destination, eg airport line, Eccles line, Altrincham line.
Don't be concerned with safety, just follow the usual streetsmarts and you will be OK.
You can pay for the tram by buying a paper ticket on the platform, or by simply tapping on and off using your credit/debit card on the yellow readers on the platforms. The system calculates your fare and you will pay the same as a paper ticket. REMEMBER to TAP OFF.
I've even walked it from the city centre to the Etihad many times, along the road and along the canal. Takes 30- 40 minutes.