Tablet turning off and on.

So did the custom ROM install successfully and failed to boot or did it boot up and then just went into a boot loop? Also is the tablet still recognized by adb when plugged into your PC?

Hate to say it but I would recommend trying to return or exchange the battery you ordered if it is not to late. Most good replacement batteries should have around a 40% charge or possibly lower but should be able to power on the tablet without issues. Most are not aware that most lithium ion batteries in electronic devices such as this actually contain a protective circuit that is used to protect the battery from damage. The battery can actually enter a protective sleep like state if the battery has sat for an extended period of time causing the battery to discharge below 3v or lower. When in this state the battery appears to be dead. In reality it just needs what is called a boost charge. Kinda the same principle as jumping a car battery. You are supplying a larger current for a short period of time which brings the battery out of the protective sleep state. Then the battery can continue being charged on its original charger. It can be hard to find boost chargers and they can be pricey. So this is why I would try returning the battery you bought if possible just for the fact it has probably been stored in poor conditions (too hot or cold of a warehouse) or has had a long shelf life which may have triggered the protective circuit. If you can not return it at least you know that you may be able to boost charge it to get it to charge and function normally.

The reason I am thinking a battery issue is the fact that when you on rare occasions can enter into the recovery menu it shuts off shortly after. Also the fact it only makes it to the Samsung logo when trying to boot it up. Your battery may have also been the cause of the custom ROM not booting and entering a boot loop. Your battery likely has enough power to turn the device on but not enough power to boot the operating system. It requires more voltage to boot the operating system than what is required to simply power on the device. Possibly the same applies to entering into recovery or download mode. That may be why you're getting stuck at the Samsung logo and also having issues in recovery and download mode. I know a common cause of boot loop can be from a failing battery as well as many other things.

So I highly recommend trying a different battery or trying to boost charge the one you recently purchased if possible to see if that will resolve the issue. Hope this is helpful.