The ideal time to plant a pine tree is early spring or fall. However, considering the cooler temps and rain we received yesterday, you should be ok to plant now. I think that is safer than leaving your trees potted until fall. Your newly planted trees must be kept watered (but not drowning or flooded) for the first couple of years; it can take up to 3 years to become firmly established.
Choose a sunny site in moist, well-drained, slightly acidic soil. If you do not know your soil's pH level, I recommend you conduct a soil test. Self-mailer soil test kits from the Michigan State University Plant and Pest Diagnostic Lab are available. You can obtain them at the link below. Full instructions are there and also come with the kits. The cost is $25. The Lab will make recommendations for any needed soil amendments. Be sure to tell them you are planting white pine trees.
When planting container-grown trees in well-drained soils, dig a hole that is 2 to 3 times wider than the diameter of the tree's rootball. The depth of the hole should be 2 or 3 inches less than the height of the rootball. If the tree has been in the pot most or all of it's life, it may be root-bound, meaning the root ball will have more roots than soil and you will probably see girdling roots growing around the sides of the pot. You will need to gently separate the roots in that case, and cut any girdling roots before planting or they will continue to grow around the tree eventually choking it. Do not completely break up the root ball at planting time, but do cut any circling roots prior to planting. The tighter the root ball, the more the root system should be disturbed.
On the other hand, if the tree has not been in the pot very long, its roots may not yet be established enough to hold the soil/root ball together. In that case, set the pot in a semi-shady spot and keep it well watered and allow the tree to continue to grow a bit before planting.
You can back fill the hole with the same soil you dug from it. ; soil amendments should not be needed. Tamp the soil firmly to seat the tree in place. It will likely sink a bit as the ground settles, but be sure it does not end up planted too deep or too shallow. The link below will give you additional planting instructions.
Yes, you should be concerned about your neighbors ailing white pines. You should encourage them to find out what is wrong with their trees as there is a potential for disease or insects to move from their trees to yours. Also, planting your tree in soil containing tree roots from a removed tree will definitely affect your new tree. Depending on how old the roots are, they could still be taking up nutrients and moisture from the soil, creating a lot of competition for your new tree. You should remove them if possible.
There are numerous insects and diseases that can affect white pine trees. The links below discuss some of the common insect and disease problems common in white pines in Michigan.