Please accept my appology for the delay in respoding to this post, but it has a number of complex issues that has required some careful thought before I could begin to formulate an answer.
1.-"We hear a high-pitch whistle from the water heater when both toilets are flushed, OR when faucet water is running in the sink at the same time"
A high pitch whistling sound can only occur in a water line when the water is in motion. While you may be hearing the sound in the close proximity of the water heater we can rule out the water heater itself because the noise occurs when the toilets are flushed and they are only connected to the cold water supply, thus there would be no motion of the water through the water heater.
The whistling sound is usually a direct result of excessive velocity of flow, which then indicates an obstruction in the water line. One of the most common causes is the improper use of a throttling type valve (globe valve) as a zone valve instead of a "full bore" valve such as a gate valve or ball valve. It can also be caused by a gate valve that is not fully open or it could be caused by a substantial dent in the pipe wall. I would begin by checking all valves in that vicinity to make sure they are fully open then inspect the pipes for physical damage.
2.The pipes seem to jump when the faucets are turned off
This is a primary indication of "Water Hammer". Normally water hammer can be corrected by turning the water supply off at the main, then draining the whole house water distribution system. Once the system is drained you can turn the main water supply valve on again and as the system fills the water hammer arrestors will be recharged with air as the pipes fill. This house is only 9 years old so it may not have field manufactured whole house water hammer arrestors but may rather have smaller commercially manufactured water hammer arrestors located near all the fast acting valves in the house such as at the washing machine supply, dishwasher supply and on the watercloset supply lines. If this is the case it may require changing the water hammer arrestors.
3."-We use to take lots of baths, but can't get enough hot water any more, so we have to take quick showers
-Shower water won't stay hot (daily) longer than 7 or 8 minutes"
The post does not mention whether this is an gas or electric water heater so let us examine some of the problems associated with both types.
If this is an electric water heater these symptoms are usually indicating that one of the heating elements or one of the control thermostats have burned out. (If this is an electric water heater please repost and I will offer you the procedures to diagnose and replace the thermostats and heating elements.)
If this is a gas water heater these symptoms normally indicate an excessive amount of sediment buildup in the bottom of the tank. As the sediments build up they form a heavy mud in the bottom and in a worst case scenario that mud is then cooked into a hard ceramic like paste that insulates the heat from the burner away from the water thus greatly reducing the water heaters ability to heat the water in the normal recovery time. You may have some success with flushing the tank but if we wait too long to flush the tanks quite often the only solution is to replace the tank. (normally the tank should be flushed at least once or twice a year and in areas where you have a very high mineral content in the water i would recommend quarterly flushes.)
4."-Tub has a slow draining clog that won't go away (or rather returns) no matter what brand of septic-safe drain cleaner used (Liquid-Plumr, Drano Foamer, Max Gel, etc)"
At best chemical drain openers are an iffy proposition. The often will work if the clog is confined to the drain trap where the liquid is held in place long enough to dissolve whatever is causing the clog, but once liquid drain openers go into the pipe when they contact a clog they will usually only open a small passage just large enough for the remaining liquid to flow past, leaving the remainder of the clog to build up again in a short time. The only effective way to clean a drain line is by means of a powered drain auger that is equiped with a cutter head matched to the diameter of the drain line.
5. -Toilets won't flush during heavy rains (BIG PROBLEM)
-Also once begins to flush and the other will continue to bubble/gurlge/percolate (and still won't flush)
-Spin cycle doesn't drain all the water from our new washer (sometimes; especialy during the rain)
The key here is that this problem is associated with a rain. I would suspect that there is a sump pump or perhaps a downspout drain that is improperly connected into the house drain, waste and vent system. (the Plumbing code prohibits connecting a sump pump or storm drain to a house sewer line) During a rain it then innundates the septic tank with excess water which results in the drains backing up until the septic tank has had time to leach the excess water into the soil. This could also be a result of a leachfield that has a marginal leaching ability and when the soil is saturated with storm water the leach field is temporarily rendered inoperative. One solution might be to increase the size of the leach field. There is another option that is rapidly gaining approval in most jurisdictions although it is rather expensive They now add an aereation chamber onto the septic tank and the water from the tank can then be routed directly to a storm drain or nearby watershed.
6.Occasional odor comes from under kitchen sink; but it fills the front of the house quickly (this was most evident after the septic tank was pumped out)
I would suspect you may have an air inlet valve serving as an auxillary vent under your kitchen sink, and if so, the plunger inside the vent may be occassionally sticking in the open position. If so, it is a very easy task to change the inlet valve.