houndlove gave you some great advice, and I'm going to repeat some of that.
You've got a couple of things that you (and your family) really need to understand. First, you have a baby there. At four months old he's like a 1 year old child (roughly) and he can't be expected to understand consequences for actions at this point. He doesn't poop on your bed because he's bad or being spiteful. He's just being a puppy, an untrained baby, and he can't help it. If you tell your family that he's no more advanced than a human child of a year old, it may help them understand that he's just not able to be "perfectly trained".
It has nothing to do with control. A four month old pup is not trying to control you, nor is he testing the waters. He's just being a puppy. If he potties in the house, that's 100% the fault of the humans around him. He can't understand fully, at this age, that he needs to only potty outside and when the urge hits him (or when he gets anxious or stressed) he's likely to go wherever he happens to be - or go to a place where he's pottied before in the house. This is NORMAL. The way to deal with it is to not let him run around the house when you can't actively be watching him. That means your eyes need to be focused on him, and not on the TV or the stove or the shower. When you can't watch him, he should be confined in a small comfortable area.
I have a 4.5 month old pup right now and I've had her for over 2 months. I don't have a problem with her housebreaking because I keep her confined when I can't watch her. When I take her out of her kennel area, we run straight outside where I stay with her while she pees and poops - and I praise like crazy when she does that. She comes running back to me with her ears forward and her body wiggling because she's learned that she's done something to make me happy. Then we go inside and she gets some free time to run around, interact with the other dogs, etc. while I WATCH her. If I need to get some work done or take a shower or anything else, then I fill her kong with goodies and I put her back into her kennel (actually she races ahead of me and has learned to smack the kennel door with her paw to get it to pop open so she can go in .. *L*). She likes her kennel because I feed her there and I make it a good place for her - and I make an effort to spend good quality time with her frequently every day.
Keep in mind, too, that you've got a breed (like houndlove explained) that is not going to just do whatever you want just because you want it. Beagles are independent dogs and have been bred to think for themselves. I have chows and they have pretty much the same attitude. I have to be very consistent, fair, kind and rewarding to make them work for me and it turns out really well, but it does take some effort on my part. My shepherds pretty much ask me "what do you want us to do for you??" while the chows are more like "what's in it for us?". You've chosen a breed that's going to take a lot more effort on your part than the herding breed mix you also have. You can't compare the two.
YOU need to become a better trainer for this little guy - if you or your family think of him as "bad" or as deliberately doing bad things, it's not fair to the dog. He's a baby and he needs to be treated that way. And you need to learn how to work with an independent breed. You can do it and in the process you'll learn a LOT (and that's a really good thing - I wouldn't be nearly as good of a trainer if I only worked with easier breeds!).
Good luck, love the little guy and please have your family read the information you're given here so that they can understand that the puppy is just a baby too.
Melanie and the gang in Alaska