May I suggest a Lands End coat? I just got one of their coldest-rated knee-length quilted down jackets, and I'm so toasty now when I walk the dogs, it's great. Plus I wear really warm mittens and wool-blend socks and warm lace-up boots. They make coats for beagles, too. I'm in NC, so it doesn't get so frigid here, and we enjoy our winter walks as long as I'm dressed right.

I used to have to walk my dog in Michigan winters and swore I was going to buy one of those one piece zip up snow mobile suits to make it both warmer and easier (you can pull those on over your pj's for that last before bedtime walk).
I never found one in my size that didn't cost a fortune but that was before ebay, craigslist etc. Give it try? I did buy some pretty high end polar fleece hoods with the face mask (not all polar fleece is created equal) so that I looked like a burgler, because we used to walk in a very windy field. Occasionally I would break out the ski goggles too.

Oh this is a big topic around our house this winter with our border collie. Everything for him has to be challenging AND active. If it is too easy, he's not interested. It's not easy to tire him out in the best of conditions! Here's some of things we've been trying.

Outside:
1. Invest in good winter gear for you, but also buy your pup a doggy backpack. You can add weights to help tire him out.

2. Get the most bang for your time outside: We used to walk and run daily. Now its all running. We also take the more challenging paths: uphill, steps, wooded terrain and he wears his backpack. :) Course if the path is snowy/slippery, take care but add weight to the backpack.

3. Anything outside of the house helps: Classes, dog parks, rides in the car. We also go the parks and play around the kid's jungle gym -kind of like improvised agility. :o We're enrolled in agility classes for Jan. Classes are a great way to bond with your dog AND tire him out.

Inside:

1. Lots of obedience work and trick practice to occupy his mind. We do drills while making meals, watching tv, when ever. There are a 1000 ways to incorporate games into everyday activities -even mopping the floor has become a new way for him practice various commands. We invested in a great book of tricks for Christmas. Sometimes I just can't think of anything new to teach him. :)

2. If you have stairs (and are in a house or have tolerant neighbors), playing fetch up/down stairs really works wonders. 'Course common sense disclaimers applies: not for puppies or elderly dogs or dogs with known joint, back, hip issues; not for dogs with long backs (doxies, corgies, etc); not recommended on wood or slippery steps, etc. etc. etc.

3. Indoor "agility": It isn't all about running top speed and jumping. The key to agility is team work. Have him go under (and/or over) a chair/card table/box, weave a course of chairs/speakers/boxes, etc. Great creative. He'll have to use his mind and body and he'll get practice working with you in a new way.

Does your pup play fetch? We do a lot of that indoors when it's chilly out. If he doesn't fetch tennis balls try squeaky stuffed toys - those are a big hit at our house.

You can also get food-dispensing toys. We have the Tricky Treat Ball (which I like because it's rubbery and not plastic, which can be noisy when dropped). You can put your dog's daily ration of kibble in it, a handful at a time, and let him roll it around on the floor till the kibbles fall out. If he's like most beagles he'll work till he gets every last piece of food out! If he likes this you don't need to feed him from his bowl, just use the toy.

There are also soft puzzle toys like the IQube or similar toys that require pulling smaller toys out. The trick is not necessarily to physically tire him out, but to keep him occupied and tire his mind as well.

Just wanted to add that for those considering the doggie backpack idea, you need to condition your dog to wearing the pack empty first, and then slowly adding weight to it. Just like with human exercise, you don't want to do a lot of weight right off the bad - it could cause an injury.

I suggest you look for a clean empty parking lot in a better part of town. I use to live in the mountains of Northern California where the snow fall can reach 4 feet in a 24 hour period. We ran our dogs in the parking lot of a nearby park which was usually plowed on a regular basis. Running into the warm car every 15minutes to warm back up and then out again for some more laps around the parking lot....good luck

In addition to what was mentioned there are other games. Like hide & seek, find the toy (or person), tug, fetch if you don't have a lot of fragiles.

Seeing that your breed excells at tracking, you might look into that. Even though that is an outdoor activity, I wouldn't be surprised if you could imagineer a way to do it indoors.