It is possible for one person or entity to rent several VPSs and run a full node that runs Core 13.1 (or some other version of Bitcoin), so while counts of various types of nodes does mean something, it doesn't necessarily reflect the overall support of the community.
I am really looking for the opinions of major Bitcoin companies such as Coinbase, BipPay, Circle, major exchanges, gambling sites, and others. Do they wish for SegWit to be implemented, do they wish for SegWit to not be implemented, are they neutral?
It would not be smart to not implement SegWit if/when it becomes part of the mainnet.
Why? Segwit brings many improvements to Bitcoin that we need, including scaling. Many wallets have or are in the process of implementing segwit.
I agree that SegWit allows for a greater number of transactions in a block, but I don't think it is the most optimal way to achieve this. I am not sure how serious other problems that SegWit "fixes" really are for Bitcoin, especially malleability
Wallet software is being changed to implement SegWit because the creators of such software believe that SegWit will likely be implemented, not necessarily because they believe SegWit is what is best for Bitcoin. If it was apparent that Bitcoin was about to HF to have a maximum block size of 2 MB, then wallet providers would likely also be in the process of upgrading their software to support this.
Some Bitcoin companies may however not "support" SegWit and would prefer that it not be implemented. My question is, how can a non-mining entity that is part of the Bitcoin economy and/or ecosystem show their support/non-support of SegWit?
Nodes that signal segwit have the NODE_WITNESS service bit set. So to check if a node supports segwit, just check for that service bit.
Even if I know that a particular node is run by coinbase (for example), the fact that this node 'supports' SegWit, does not necessarily mean that coinbase supports SegWit. If SegWit does in fact get implemented into the Bitcoin network, then coinbase (in this example) will want to have prepared for, and tested SegWit sufficiently.
Even if an entity does not support SegWit they may prepare for SegWit in the event that it becomes part of Bitcoin despite their wishes.
Segwit is backwards compatible. If you don't like segwit, you can continue to use Bitcoin as you do now with almost zero impact on you.
My understanding is that if a node does not upgrade, then it does not have a way to validate that a transaction is valid (due to being unable to validate the transactions signature), and does not have any way to validate that a block that contains one or more SegWit transactions is valid. Am I right about this, or is this not accurate?
If the above is accurate, then I think you are using a very specific definition of "backwards compatible" to make your statement.