Best distribution of time will be affected by how many day-trips you plan in each area as well as sights of interest in your base city.
Nice has many art museums. Check your guide book to figure which, if any, are of interest. If you aren't an art person, you may primarily want to see the (small) historic area and go to the market, then spend the rest of your Rivera time on a series of day-trips. Which day-trip destinations you choose will again depend on your interests. Menton, for example, has two really nice gardens, but its impressive historic district tends to be packed with tourists. Again, consult your guide book.
The regional train makes reaching just about any coastal spot quite easy from any other coastal spot. Its relatively central position makes Nice a convenient base. That city is also a hub for buses up into the hills, including the bus to St.-Paul-de-Vence. You could easily fill a week in Nice with six days of side-trips, but that wouldn't leave you enough time for the two other areas you want to visit.
I enjoyed staying in Avignon a lot. If you step off the direct path from the train station to the Palais des Papes, you'll see far fewer tourists. In this area you'll probably end up making some side-trips by bus as well as some by train. You may want to consider a one-day bus tour in order to reach several of the smaller places that would be difficult or impossible by public bus.
However, a lot of the bus tours go to places you might conveniently reach on your own, so here you may find consulting Rome2Rio.com helpful. You must ignore travel times, frequencies of trains/buses and fares, because that information is very unreliable. But the website will be helpful in indicating whether you can get from Avignon to Town X by train, by bus, or by some (probably awkward) combination of the two. It is also usually accurate about the transfer points it shows. This will give you a good idea of whether you would want to use a tour to get to a particular spot. When you want to get into the nitty-gritty of a potential side-trip, keep drilling down on Rome2Rio until you find a link to the appropriate bus company; go to that website for schedule information. For trains, go directly to the SNCF website.
Note: April is very off-season for the Riviera and Provence. It will affect frequency of transportation, especially bus transportation. Be sure you're looking at the correct schedule. Many sights will have more limited hours in April; check their websites.
From Toulouse I recommend a side-trip to Albi. The Toulouse-Lautrec Museum is good, and the cathedral is grand. During the siesta period in mid-summer (which may have been a factor), Albi was pretty lifeless, though handsome. That was not an issue in Toulouse.
I highly recommend taking the full 14 days for this trip. With only 7 days I absolutely would not try to travel to Toulouse. Even with 10 days I would stick to Nice and Avignon, because there are so many worthwhile things to see in both of those areas. Depending on public transportation in Provence will slow you down quite a bit. But I am a card-carrying slow traveler.
Part or all of your Nice-Avignon and Avignon-Toulouse train trips may be on TGVs. Tickets on those express trains can be bargains if you buy a non-exchangeable/non-refundable when your travel date first goes on sale, but you'll pay a lot more if you buy shortly before the day of departure. So if you get things locked down early, it's worth looking at the SNCF website (above) to see if there are some bargains available that work for you. I took just a quick look and found a price range of about €26 - €90 for Avignon-Toulouse, but it's possible fares can go lower or higher than that. Do pay attention to travel times, because there are variations in how many transfer you need to make and where they are. The most expensive route may not be the fastest.