To quickly answer your questions: Yes, Swan will transfer your luggage. No, it is very inefficient to walk to Epcot and then catch the monorail to MK. Yes, getting strollers on buses is very manageable, but it is much easier to just stroll them onto the monorail. For the detail behind this advice please read below which is quite similar to your trip.
We took our first family trip in 2003 with our 4 yr old DD. First and foremost, by deciding to stay on site you have already made a smart decision. It sounds like you may be like us in that you can afford to stay at the monorail resorts but still find it hard to choke down the added $100-150 per night expense. We almost stayed off site at what looked like a very nice hotel but based on advice from message boards decided to stay at the Swan which was very easily affordable with the teachers rate and seemed too good to pass up. We had read that Swan is not quite the Disney experience as a true Disney resort so we decided to split our stay with the CBR.
First, Swan is very nice and we loved it. The move to CBR mid-week was very uneventful and easily managed, but it definitely felt like a downward move we probably should have stayed there first instead of the Swan. That being said, our trip was still fantastic so much so we decided that WDW would be an annual family trip. That being said, after eating dinner at Chef Mickeys and the Kona Café we decided we definitely wanted to stay at either CR or the Poly on our next trip knowing full well that even with AP or other discounts it would add at least $700 to the trip ($100 per night for 7 nights), or possibly even over $1000. Believe me, it still hurts to book that, if nothing else due to our upbringing as no matter our household income $1,000 will always be considered a lot of money in our family.
We booked CR Tower for our 2004 trip (like you, we took my parents this time see more on that a few paragraphs below) and with the Disney Visa deal they had going we were able to get it for basically $250 per night. Still a great deal and we absolutely loved it. Yes, CR needs a rehab but some make it sound like a motel 6 with a monorail. We dont sweat décor at all, and in many respects it was nicer than the recently refurbished Swan room (e.g., CR rooms have a separate toilet and vanity room from the main bath). Swan beds are indeed heavenly and I doubt any Disney resort can match them. But to us, waking up with a view of the castle, watching fireworks from the balcony, extra convenience of the monorail and the easy walking alternative to MK (probably 5 minutes with my parents although both are very ambulatory), it was definitely worth it. Another added bonus is the ease to which you can zip back to your room for afternoon naps, or perhaps in your case, just you and the infant while the older kids stay with your DH. Also, if you and DH want to grab a nice late dinner at GF or Poly (we loved Kona Café), it is so nice to not have to worry about linking transportation via DTD back to your hotel (although obviously Epcot will not be a problem if you stay at the Swan).
We are now planning our 2005 trip and like you, will be bringing an infant this trip (will not be bringing my parents, however, God love them). Unfortunately, we have been unable to get a CR-Tower room (even at rack rate we are guessing it is scheduled for rehab) and thus we have opted for the Poly.
If we had to do 2003 over again, and in fact we thought this during that trip, we wish we had moved from the Swan to CR or Poly and not to CBR. Although to look at the silver lining were glad we did the Swan to CBR if only to give us a good comparison and now we know for sure that we prefer the monorail resorts at least while the kids are younger and that the added expense is worth it to us. At a minimum I would suggest Swan to CR-Tower (or Poly as second choice) that way you have a nice comparison that shouldnt leave you with too many regrets either way. Also, I would do Swan to CR as I recall going down from Swan to CBR was kind of a bummer. Of course, I suppose going Swan to CR could be bad if you build up expectations too much.
Another issue to note is that your parents may think very differently from you, which may or may not be a consideration. While mine both grew up middle class each of their parents (my grandparents) grew up relatively poor (although during the great depression they were probably still middle class themselves if you know what I mean), so consequently both my parents are kind of cheap when it comes to travel. While my parents can easily afford the deluxe resorts and find them nice and convenient, they still have sticker shock (part of that is they are just not used to traveling nowadays and really have no idea how much things cost) but if your parents are like mine, I could see them not fully appreciating the monorail resorts and thinking the Swan or even the All Stars are just as nice.
For example, if your Dad invites the family out to eat at a relatively decent restaurant something above Chilis perhaps but not much so and thus he knows the meal will likely push $100 (in our case, 4 adults, one child), and yet he still feels compelled to act incredulous toward the waitress at the price of a glass of wine before he orders it, then your Dad may be like my Dad!
In short, I say go with the monorail, and in particular CR-Tower. If you want to ease your way into the monorail resorts especially if your parents are thrifty like mine then get the Swan teacher rate and do a split stay. If an extra $500-1,000 for rooms would really hurt your vacation budget (and not just be hard to pay due to upbringing, etc.), then go with the Swan teacher rate as it is probably the best bang for the buck. But I sincerely believe with your travel party, the CR is the way to go.
Sorry to ramble, but I thought the insight would help provide a basis for the advice and better help your decision.
We're trying to decide which resorts will have the easiest modes of transportation with the baby in tow. Is the contemporary worth the price? Does the monorail take you to all the parks from there or just the MK? Are we better off to stay at the S/D near Epcot & hop the monorail back & forth to the MK? 
I'm a teacher & have heard they have pretty decent discounts. If we would chose a moderate for better price is the bus transportation going to be a problem with a baby/stroller, etc.? Thanks for any help you can offer. 
in 1992! We stayed at the old Dixie Landings & loved it!
