Co.A uses a trademark in its name but does not register in the Fed Register but Co.B gets a federal registration of the same brand and has the official R can it force Co. A to cease using the name?

If company A used the trademark on its products before company B did, then it doesn't matter if B later gets a registration; they cannot stop someone who had used the same brand earlier. However, if company B had been using the brand longer, but never bothered to register it until A came along, then B has the superior rights. Trademark rights start from the moment you first use the brand in association with marketing your products or your services. You can also obtain provisional protection by filing a federal application for "intent to use" a trademark. Anyone who starts using a similar mark after your filing date can be forced to stop, once your registration is granted. Sometimes competing brands interfere with each other during or after the registration process. If you believe someone is registering a similar brand in a way that will harm you, there is a mechanism to "oppose" an application or even "cancel" the registration, since it should not have been issued. Alternatively, there is a way to have "concurrent use" of the same trademark by different owners.