Hi vicemage,
I hope by now you have talked to your vet. The most important concern is if your cat has not urinated. Male cats can become blocked because they have a narrow urethra. If it becomes blocked it can be life threatening very quickly. You mention your vet "induced" urination. How was that done ... was your cat blocked before and the vet inserted a catheter (tube) to unblock him? If so, then he would be at a greater risk of another blockage which is an emergency situation.
At 10 yrs old, there is a greater possibility that the problem is a bacterial infection. Usually the specialists give about 10 yrs as the turning point ... cats less than 10 yrs are more likely NOT to have a bacterial infection (and more likely to have sterile cystitis, an inflammation of the bladder) ... and cats over ten years are more likely to have a bacterial infection, probably as a result of more dilute urine.
You are right that testing urine collected from the exam table is not "sterile" and can easily be contaminated. Generally, it is best to get a urine sample by cystocentesis (inserting a needle directly into the bladder ... which sounds bad but most cats are not bothered by it). Then they can do a "culture and sensitivity" test to see what bacteria grows and what antibiotic best treats it. The only thing a cystocentesis isn't good for is checking for blood in the urine because the act of inserting the needle can introduce a little blood into the urine sample. The problem now is that once an antibiotic is started, it will be difficult to find bacteria in a urine sample.
My recommendation (as a non-vet) is to talk to your vet (if you haven't already) and let him/her know about the lack of urination and defecation. Your vet may want to make sure he isn't blocked. The lack of eating may be a side effect of the Clavamox. I had a cat that used to get violently sick from Clavamox (i.e. vomiting). Most cats handle it well, but some get an upset stomach from it. Your vet might want to try another antibiotic.
A 10 yr old cat is a "senior". Has he had a general check-up as well? It would be wise to have a blood test (chemistry and CBC) specifically for seniors that would check things like kidney function, glucose and T4 (thyroid hormone).
In general, the best thing to do for cats with urinary issues is to try to increase the amount of water in the diet. If at all possible, feed canned food and add water to it. Also, trying different types of water bowls, running water, pet fountains, water from tuna cans, broth, ice cubes, etc. Increased water will help flush out the bladder. However, make sure that he does not have a urinary blockage first, since adding water to a blocked cat could be fatal.
One last suggestion is environmental ... to reduce stress as much as possible. There has been a lot of work done at Ohio State by Dr Tony Buffington that shows the connection between stress and urinary problems in cats. OSU has created a website called the "Indoor Pet Initiative" with many helpful suggestions on ways to enrich a cat's environment. They have found a great success rate at reducing episodes of cystitis by reducing stress. Here is a link:
http://indoorpet.osu.edu/index.cfm Hope all is better by now.
Stephanie in Montreal