You can find a train time table here: www.huoche.biz/ds3d4e5f6c1c4e.aspx

It is difficult to say whether you can get a ticket the same day, it just depends and it is not uncommon that the trains are fully booked. While there are several agencies that can book a ticket for you, the only one I know that might speak a little English is the one outside Camellia Hotel. Or you could just take a taxi from the airport to Kunming station and try your luck there.

Alternative option: since you are not doing the train ride for the scenery why not just fly to Lijiang from Shanghai? There are non-stop flights and others with a stop in Kunming. Have a look at ctrip.com for prices, it might cost the same or even less than flying to Kunming.

Kunming is not super interesting as a tourist city, consider a slow return trip via Shaxi and Dali.

The border at Ruili was certainly 'open' at one point, I crossed from Myanmar into China there almost ten years ago. But it was required to be in an official 'tour', even though that consisted of just a totally useless Burmese guide and me. The reverse was also possible and popular with (mostly American) war veterans interested in the Burma road.

I heard that if you ask at the Myanmar consulate they will tell you it is impossible, but that the Camellia Hotel travel agent can get such a tour together for you. Coming from China this will most likely be a tour guide picking you up at your hotel in Ruili who will assist you in the border crossing. On the Burmese side, in Muse, you will hop into a car, which you pay for, to drive to Lashio and on to Mandalay. The drive to Mandalay takes a good day on a very good road. There is little to see though and the whole thing will cost you a few hundred dollars.

The other border crossing that is open at least to Chinese tourists seems to be north of Tengchong and connects to Myitkyina. For the appropriate fee it might be possible to set a tour up to cross there as Myitkyina is open to foreigners.

I would not think that Pianma is open as the Burmese side is probably extremely poorly connected to the rest of Burma.

It is certainly doable in seven days, but you will spend a long time on the road and you will probably spend almost as much as you would by going by plane. As far as I am aware, China Eastern has a direct KM-CM flight again.

The very quickest way to do it would be to take a sleeper bus to Mohan (the Chinese side of the Lao border) (about 13 hours), cross the border and get into one of the cars probably already waiting on the Lao side for the run to the Thai border. It will probably take you six hours from Boten (Lao side of the Chinese border) to Huay Xai (Thai border) and then another four hours or so to Chiang Mai. Making connections should not be much of problem, and with a bit of luck you could arrive in Chiang Mai the late evening after departing KM.

Luang Namtha in Laos would be a good place to break the journey, it is less than two hours from the Chinese border, so arriving by sleeper bus you could get there before lunch, and leave the next day on a bus bound for the Thai border, which would make it a two-day trip to CM.

In Mohan there are always minibusses at the bus station, which is about half a mile from the border on the eastern side of the main street. They usually do the run to Mengla, but I am pretty sure that they would do hire jobs as well. I have not seen any other vehicles for hire in town, but if you took the minibus to Mengla you would have more options.

I have always found it a bit difficult to negotiate with drivers unless you can really tell them where you want to go, because only if they can estimate the distance they can give you a price.

However, my rule of thumb is that in a small minibus or a car the price for hire works out as RMB 3 per kilometer. Any waiting time should not really cost extra as drivers tend to spend all day waiting anyway, but you need to tell the driver, otherwise they get impatient.

There is at least one daytime bus daily, leaving the western station 西部汽车客运站 at 9am, arriving some 10-11 hours later. I have been on this bus once, and even with the bus being pretty good it was quite a tiring ride, but then it is one of the longest journeys in the province. The bus makes stops for toilet and lunch, but does not waste any time by pulling into Dali or Lijiang.
There is also about a dozen sleeper busses going overnight, but beware of thieves. (A friend of mine reported that his bag had been tampered with, but nothing stolen on this journey overnight.)
Alternative option, as mentioned before: sleeper train to Lijiang and bus from there.