The answer to all of those is "yes/no - because (network) consensus"
1. It's entirely possible to change the reward halving schedule, or even remove halving, in a new set of rules - but unless you get the majority of the network to agree to those new rules, any blocks you generate that go against their rules will simply be rejected by them.
2. Older clients won't work because A. if they're used to mine against, they would be creating invalid blocks (see above) or B. if they're used for verification they simply won't know what to do with transactions/blocks that use the newer rules or find them in violation of the older rules, and won't accept and/or relay them. If the majority of the network has updated, that effectively means your client will be stuck in the past.
3. There's nothing preventing that, other than the network consensus. The network - whether you view that as miners, nodes, services - isn't likely to adopt such a change without some very serious discussion and agreement overall.
You can see this sort of thing with the recent block size change proposals. There's nothing to stop people from making software that has different rules, like XT with BIP101, but (in this case) any blocks generated that break with existing rules (a block larger than 1MB) would get rejected. That's why most of these proposals introduce a method by which a minimum portion of the network needs to have indicated support for the change (what you're alluding to with "95% of the last 1000 blocks", although those values can be chosen differently and arbitrarily) before the change actually goes into effect
for those clients.
If a good chunk of the network still keeps with existing rules then you end up with forks of the block chain, like what happened a while back with v2 vs v3 blocks.. some miners had indicated they were all set for v3 blocks, but when the change went into effect they actually tripped all over themselves and the block chain forked for a few blocks until those miners woke up and smelled the coffee.
The block size is a highly politicized but ideologically fairly minor change and it's proving very difficult to find consensus on that issue so far - imagine trying to push through any change that would actually touch Bitcoin's supply characteristics
"therealsteve". I found your answers very informative. Thank you very much for taking the time to answer my questions in such great detail.