I'd try to eliminate the flexible hose if at all possible. You can get a threaded piece of copper pipe and solder it into your plumbing system if you have copper plumbing.
In a well designed system, all of the hoses, valves, etc should exceed the threshold for the TPR valve on the hot water heater. So maybe the hose you had in there wasn't rated for enough pressure, or your TPR valve isn't functioning (very dangerous, you should test it).
For the tank itself, don't worry about water coming out of it. It will work just fine without water in it (in fact, it will likely work better since you've increased the volume of air absorbing pressure fluctuations). To pressurize it, you have to remove all water pressure first (shutoff the water supply and open a faucet) and then pressurize it with a pump up to your water pressure PSI level. As long as the tank holds pressure with the water turned off, the bladder is still good, and you can reuse the same tank.
If you're worried about dirt in the water side of the tank, fill and drain it a few times to see if you can clean it out. Chunks of rubber will likely get stuck in an aerator or shower head which would need to be cleaned out, so better to do this now.
In terms of mounting, they can be mounted any direction. If mounted vertically on a fixed pipe (doesn't matter which end is up) then you don't even need support brackets.
That sounds like an overly complicated setup with far more pumps than it should need. Or else you said pump when you meant tank in two places, actually, re-reading it.
In short, you can have as many pressure tanks as you like. Locating them closer to the point of use (the house) would be helpful, to a limited extent. That limited extent is the "effective volume" of the pressure tank at the house - while it's got pressure, you get better pressure at the house. As soon as it runs out of water and you become dependent on the pump, the pressure loss of the 300 feet of 1" PVC to the house comes into play, until you slow down your water use to the point where the pressure tank at the house fills up again.
If you do not change the setpoint pressure, the peak water pressure in the house will not be affected at all. Once the pressure tank at the house is depleted, the pressure will be the same as it is now. If you have a pressure gauge at the house now (or can add one) it might be useful to know the pressure at the house when "loads of interest" (showers, say) are drawing water.
Most pump controls can be altered to have a higher setpoint pressure, and most well pumps will provide that, within reason. General recommendations are to stay below 80 PSI - but if you are now running your pump on at 20 off at 40 (a typical default) you could turn it up to on at 30 off at 50, or on at 40 off at 60 - you will need to adjust your pressure tank (and any you may choose to add) for the higher pressure, particularly if they are bladder-type tanks - with the system drained, a bladder tank should typically be 2 PSI less than the low water pressure setpoint (ie, 18, 28, or 38 for the three ranges I've just given) and you may need additional pressure tank volume (because the effective volume of water a pressure tank can hold goes down as the system pressure goes up.) So, you can probably get more pressure without another pump, and possibly without another tank, but we'd need more details of what your system is doing now to know that for certain.
Unless your water use is extreme, 300 feet of 1" PVC pipe should not have a lot of effect on the pressure - at 5 gallons per minute, about 2.2 PSI - at 10GPM, 8.2PSI, 15 GPM, 17.4 psi
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You have the idea of it, I think, so long as you're not going for something cockamaimie like tying the air charge sides together. I don't actually think you are saying that; in any case, don't do that.
You just set those both 2 psi below the cut-in (ie, 18 PSI for a 20/40, 28 PSI for a 30/50 38 PSI for a 40/60 pressure switch setting, with the water side empty) and connect the water line to both tanks.
If the old vertical tank is still in usable condition, you could tap it in as well, though logic would call for a 1/4 turn ball valve; that might be a good idea for the new tank as well, since eventually it will be old, and may need service/replacement - price of the valve .vs. being able to service/replace the tank without shutting the whole system down - your call.