Hes 18 level 10 and I'm not able to show him. Is he retired? Is it because I'm only able to get level 9 riders? Has he been shown 3 times this week? I dont know how to tell these things

If it's the rider then how do I get a level 10 rider for him? I'm trying to collect a few nice show gelding for myself

Also when does the new "year" start? Is it monthly?

Once a horse reaches lv 10, it retires and won't be able to show in that discipline. However they can be used to give a rider an extra training boost once per month. For that go to stable-->training centre and scroll down the page to where you can train your riders. There you'll get the option to train them with a schoolmaster or to do just the normal training.

If you want to know how often your horse has been shown, you go to "Manage your horse" and click on "Enter Show". There you'll be able to see how many times your horse has been shown. Alternatively you can go to your barn, scroll down, go to Sort/View and change the view to "Showing". Then you'll be shown all horses that haven't been shown 3 times this week.

If you're looking for show geldings, I'd start with lv 3-5 as they're fairly cheap (below 5 k) and they still give you a nice income and won't retire too soon. So on the long term you get more money from them than from the high level geldings that will retire soon. Also a horse has a chance of dying each week once it's 18 years old. This chance gets higher the older your horse gets. So if you're really unlucky you could end up buying a horse that passes away after a few days.

A game years is one month in real life. The new years starts at 2:30 am.

I can also advice you to have a look into the guides: https://horseeden.com/forums.php?f=11

there are a couple of really good ones about showing and getting started in this game. They helped me a lot.