The 7x14 lathes can be pretty good little lathes from what I've heard, but they take some work. The main issue being that they are about the cheapest lathes available so they make a lot of cost cuts to meet their budget.
Basically what you get by buying a more expensive one from a place like Little Machine Shop or Micromark is they have already done many of the improvements that people often make to them so you don't have to. I've had good luck with Little Machine Shop for other items, never had any issues with them. They also sell an extended length 7x16.
Once you get into the $1200-1500 range it seems like there is less need to rebuild them, so the higher end 7x14, 7x16, as well as the 8x16, and 9x20. This doesn't mean there is not room for more improvement, but you should be able to start working without fixing up the lathe.
The 9x20 lathes are about the largest that I would consider remotely portable, and at 250lbs portable meaning bring a friend or get an engine hoist. Unless you are a weight lifter you will not be moving it by yourself. The 10x22 lathes weigh about 100lbs more.
You mentioned in your other post you don't have a large space. If you plan on moving the lathe out of the way when not in use, here are some approximate weights to consider.
Sherline or Taig under 40lbs
7x14,17x16 100lbs
8x16 160lbs
9x20 250lbs
10x22 300+lbs
In length you are looking at needing about 3 feet up to the 9x20, the 10x22 needs about 4 feet.
You mention wanting to turn something 4" in diameter and a length of 16" between centers.
The 4" diameter should not be much of an issue for the 7" and larger lathes, but if you actually mean a part 16" long, you will need more than 16" between centers. When a lathe says 7x16 that is the theoretical maximum, but reality gets in the way.
You will not be able to turn a 7x16" cylinder, because (based on the LMS 7350 specs) you can only turn something 4.3" over the saddle. As far as length, "between centers", means between centers, two sharp points that hold the ends of the part, no chuck. Awkwardly for anything too large to fit through the spindle (bigger than 20mm) you will not be able to drill the ends of a 16" piece which is required to hold it between centers.
If you need to maintain some mobility, then one of the LMS 7x16 lathes is probably your best bet. If you actually need to be able to work on a 16" long part, then you will not want to go with anything smaller than a 9x20 or 10x22.
Welcome to the fun, there is a lot to absorb when you first step foot into this stuff but if you like making things it is worthwhile. The lathe is a core tool, so don't short change yourself. I have a Sherline and think it is a great little lathe, and I'd love to be able to recommend one, but it really is too small for the projects you are describing.
For the kind of things you are looking at I don't think you would be disappointed with a 10x22. The PM 1022 is well regarded but is pushing your budget. Look at the tooling a lathe comes with when you are making comparisons, the PM1022 is one of the more expensive lathes in this size but it also comes with a lot of good quality tooling, more than most.
PM is often brought up for its good customer support. Grizzly is sometimes lumped into the "cheap" Chinese like Harbor Freight or random ebay sellers but they are a reputable dealer with a long history. They are a big company so probably lack some of the personal attention you get from a smaller company like Sherline, LMS or PM, but I've had nothing but competent and responsive service when I have bought an item from them. If one of their machines is a better fit to your space or budget, I would not worry about getting support if you have an issue with the machine. Most of the machines that they sell in your size range have been in production for decades.
Vintage lathes can also be a good option, but availability varies widely by location. There is also the issue of knowing what you are looking at so you don't buy a problem. This is usually an issue for people just getting into the hobby unless they know somebody who can go and look at a machine with them.
