How many cultivars - Many apple cultivars are self fertile. However, even the self fertile cultivars will set more fruit if cross pollinated by another cultivar. Wild apples/crabapples in the area can also work as pollenizer.
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Same cultivar - Using two trees of the same cultivar will not be beneficial. That would just be like having a bigger tree. You need another cultivar to get cross pollination.
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Rootball size - The figure you give is too small, especially for smaller trees. Below is my generic rootball guide. Soil type, tree species, and other factors are not taken into account, but it will give you an idea:

For trunk diameter below 1/2", multiply the diameter by 20 to get approximate rootball diameter.
For trunk diameter between 1/2" and 1", multiply the diameter by 18 to get approximate rootball diameter.
For trunk diameter between 1" and 1-1/2", multiply the diameter by 16 to get approximate rootball diameter.
For trunk diameter between 1-1/2" and 2-1/2", multiply the diameter by 14 to get approximate rootball diameter.
For trunk diameter between 2-1/2" and 4", multiply the diameter by 12 to get approximate rootball diameter.

Here are guidelines for rootball depth:

For a rootball with a diameter of 1', depth should be approximately 8".
For a rootball with a diameter of 2', depth should be approximately 1'.
For a rootball with a diameter of 3', depth should be approximately 15".
For a rootball with a diameter of 4', depth should be approximately 18".
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Leaving the rootball in place - Leaving the rootball in place for a few weeks is unlikely to help chances of survival. Root pruning can be used, but is generally done at least a few months in advance. If you dig a proper rootball, you won't need to worry about this for an apple tree.
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Linked below is a guide to transplanting trees. I think it may be helpful to your task.

Here is a link that might be useful: Planting a Tree or Shrub

Yes, there needs to be another apple tree nearby for pollination in order to fruit. It does not have to be the same variety of apple.

Generally speaking, your 10 inches for every inch of caliper is the correct guideline for digging a soilball. The only time you would cut the roots and leave it in the ground is if you need to transplant at a time that isn't ideal. You'd be better off just to get it over with all at one time in my view. I'd imagine that transplanting at this time in your location is probably just fine. Most people farther south would want to wait until later in the fall or even once the leaves drop.

how big is your tree???

you can mail order a new .. improved variety for under $50 .... so how valuable is your time ... and your back??? is it really within your physical ability????

why is this tree worth the effort...

why does it need to be moved ...

if you basically truncate all the roots... you will stress the heck out of it.. and then you have this idea.. to come back in a few weeks.. and stress the heck out of it again... no.. never ... do it once.. and be done with it.. the root pruning issue is way old school .. but if it is to be done.. it is done almost a year in advance ... not weeks ...

you should do it as soon as the leaves fall off.. or as soon as the ground thaws in spring ....

if there are crabapples in the 'hood' .... they will probably pollinate an apple ...

your best bet.. is to leave this one.. and buy the second.. ipso presto.. two trees for pollination ....

how about giving us the full facts.. before asking questions on the periphery of tree moving .... it might really help you focus on what is important ....

ken

"Yes, there needs to be another apple tree nearby for pollination in order to fruit."

Since a large number of apple cultivars (well over a hundred) are self-fertile or at least partially self-fertile, this statement is not generically true. Self-fertile and partially self-fertile cultivars can produce fruit without any other tree present.

"It does not have to be the same variety of apple."

Planting another tree of the same type as the first one will be of absolutely no benefit.

"Generally speaking, your 10 inches for every inch of caliper is the correct guideline for digging a soilball."

See my chart above or ANSI Z60.1 (American Standard for Nursery Stock). Especially for smaller trees, multiplying by 10 will result in much greater than normal transplant shock and a much lower likelihood of survival.
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Ken's statements...

As usual, Ken's asking the questions that the rest of us should have, making the rest of us look bad. LOL

sorry for not giving all the facts.. the apple tree is small about 10' tall. i prunned it back last year. I want to move it because it is obstucting the view of my pond from my window. i have a much larger apple tree about 50' away beside my house and thought i should move it near that one wondering if it needs to be close to another apple tree. they are two different types of apple trees and i have no clue what they are. LOL one has pink blossoms and the other is white!! thank you for all the advice.

Uncover the roots at end of winter and move it bare-rooted. If it was grafted on a dwarfing rootstock the root system could be quite small. Have the new hole dug or at least well started beforehand, and a plan and materials in place for staking it up after setting it in the new location. You will need somebody to hold the tree in place while it is staked and the hole filled with soil.

As always, use the same soil that came out of the hole to fill it back in, without any peat, compost etc. being mixed in.

As usual, Ken's asking the questions that the rest of us should have, making the rest of us look bad. LOL

===>>>> yeah.. its my law school training.. kinda hard to argue the ISSUE.. if we dont take the time to define the issue ...

and the issue for me.. having planted and moved hundreds of trees... and i will yell...

IT IS NOT WORTH MY TIME NOR MY EFFORT TO MOVE A 10 FOOT TREE ...

with my experience.. the odds of success are about 25%.. and with many hours of HARD physical labor.. and emotional stress.. it just isnt worth it ...

if you hire a tree spade.. that's a different issue ... but i still wouldnt invest a couple hundred dollars in moving an un-named apple ...

i will yell agin .... MAIL ORDER ANOTHER IF YOU NEED A REPLACEMENT.. AND CUT THIS ONE FLUSH TO THE GROUND.. AND APPLY ROUNDUP TO THE CAMBIAN ON THE STUMP ....

and then enjoy your pond...

it just isnt worth it...

and if you have some green goddess guilt... pshaw ... mailing a replacement leaves mother earth net zero ... and guilt goes away ...

frankly.. we can name 100's of BETTER trees to replace an apple ... with its scab.. fire blight... chemical necessities for edible fruit.. rotting dropped fruit that brings in wino bees/wasps ... i mean really ... cant you find a pink/white flower tree that doesnt have the downsides of an apple ????

ken

If you really want an apple...

Not all apples cross pollinate with each other. I would take a branch from the tree your keeping to a local fruit tree grower and see if they can identify the tree for you.

Then buy a fruit tree from them that will pollinate the tree your keeping.

I looked at apples from a local grower last month. They had many named varieties of apple trees with at least a couple dozen apples on them around 7'-8' tall for around $35 to $65.

They also had much larger apple trees that went up to about $230. You would have a substantial crop of apples from one of those large trees the first year.

I don't have experience with the mail order places, I am sure they are smaller trees than what you can find locally. However, I am sure any place Ken suggests is going to sell quality trees.

All that said, do you really want an apple tree? For like $20 we can buy more apples than we can use from a local grower. I've come to believe that most crops cost far more to grow yourself than to buy. Farmers markets are often a good deal especially if your looking for local / organic / natural.

Jamie