This is a bit offtopic and should probably have its own thread, but Ill try to address the questions anyway.
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With the current centralisation problem, is there any way to split fees from mining so that anyone can verify a transaction,
Full nodes already verify a transaction they also decide whether or not to relay a transaction if it is valid, but does not meet certain criteria. One example would be the isDust() check. If a transaction creates an output that is smaller than 546 satoshi, it is considered Dust and will not be relayed. Miners might still put this transaction in a block, its just unlikely that the transaction will reach them.
thus incentiving people to become Bitcoin nodes in order to collect transaction fees.
In order to collect the fees one would currently have to find a block. Finding a block is intentional difficult to protect the network. There is also the issue how to determine which node should get a part of the reward. Its pretty easy to set up nodes that act like they have a full copy, but actually just relay block requests. Thus your suggestion would make it profitable to set up many nodes that are not actually helping anyone, but still get payment.
OK, the mining community are not going to like this idea but...
...without them, how could you fork bitcoin? If your new fork has no mining power, its very vulnerable.
The next question which puzzles me is why, if centralised mining is becoming such a big issue, or problem, can the Major miners simple agree to
split their mining operations into smaller pieces so that they no longer encroach on the 51% threat.
IIRC that happend in the past when GHash.io was close to 51% they split up their operations.
Another question, or suggestion, why not add a small fractional percentage to the fees to be collected and then donated, either to a charity, to be used
advertising Bitcoin in the media, or as a prize for those who've done most for the Bitcoin community, or maybe all of the above.
You are free to donate something with every transaction you create to an organisation you deem worthy. I wouldnt think its wise to force an advertising budget. One of Satoshis concerns was that bitcoin might grow too fast, I think we still have some issues to resolve before bitcoin is fit for mainstream adoption.