Fly into Mandalay, exit overland at Mae sot into Thailand, and get a bus to Nong Khai(for Vientiane) or Chiang Rai (for Huay xai), depending on where you want to enter Laos.

On my first trip to Laos, I spent about 5 weeks just in the north. If you like rural life, walking through villages, maybe doing some local hikes, there's certainly enough there. Some of the highlights for me: Muang Sing and the region around there, as well as Phongsali. Both regions have been changed since I was there by Chinese investments and road construction, and I don't know if the villages there are as charming as they used to be. I also really like Vieng Xai, and the region around there.

So certainly there's enough of interest just in Laos itself (I've never been to Vietnam). At least stuff of interest to me.

"Yes, Laos doesnt really have "big must see sights""
Your definition of must see and mine must seriously differ.
Plain of Jars, Wat Phou, the landscapes of Vang Vieng, the stunning scenery in Phonsali, Luang Prabang.

"Laos doesn't have much in the way of old temples " Wat Phou is older than any of the Angkor temples.

PICTURE - Limestone karst mountains in Kasi area. Long ago this used to be dangerous area in the days when the Vientianne - Luang Prabang road was subject to banditry and armed attacks.

Get a visa in Laos. You can pay a little more to speed things up to same day or next day in Luang phabang, but in Vientiane were less obliging. I got my visa next day, pick up is 4 to 5 pm, so l did a day trip to the waterfalls. As far as I know you can get 60 days if you want it.

You can consider Vang Vieng and Luang Prabang. Vang Vieng is 150 kilometer from Vientiane to the north. 3 days in Vang Vieng are enough. For hotels, there are many types from common to luxury ones. Luang Prabang have many places to visit. Perhaps 3 days more to cover all the sights here.

You will probably need to transit via Thailand, by air or land. For example you can fly from LP to Bangkok (DMK) on Air Asia then onward to Mandalay or you can do LP to Chiang Mai (Lao Airlines) then onward to Mandalay (Bangkok Airways) the more direct but more expensive option. Cheap way is slow boat from LP to Huay Xai in Laos, cross the border, bus to Chiang Mai. If crossing into Myanmar at Mae Sai / Tachilek further overland to Mandalay has restricted road requiring a special permit. The more popular overland route is now via Mae Sot / Myawaddy to Yangon.

I dislike tours too, but for Halong Bay you obviously need one. Cat ba is the cheap option. Travelling alone I didn't want to share a cabin, or pay a supplement, or really want an overnight boat trip. My day trip was fab, and we saw Halong and Lan ha bays without any crowds. We saw few other boats, went kayaking, swam, saw a floating fishing village, Monkey island and the viewpoint, lunch, all for $20, full day. There were about 20 people on the boat. There are lots of agencies selling day trips from Cat ba town. Highly recommended if on a tight budget. You will need 2 nights on Cat ba to do a day cruise, then head for Hanoi.
The bus from Mai Chau to Haiphong won't go anywhere near Hanoi. It skirts around Hanoi. I don't know if you need to change bus anywhere, or if it is direct. You could possibly go from Mai Chau to Haiphong via Ninh binh if you had an extra day. I took a minibus transfer from Cat ba to Ninh binh (tam coc), which must also work in reverse.

I found there were a lot more transport options in Vietnam than are usually mentioned or advertised, which helped a lot with a tight itinerary and travelling alone, and avoided pointless journeys in and out of Hanoi. I booked all my onward travel through my hotel, which included a hotel pick up, so saved on taxis to the bus station.

Agree with previous posters. Stick to one country. If not, remove Luang Prabang. I love the town but just heading to Laos for one stop is wasting time you could be exploring elsewhere. Battambang could be missed also.You'll need minimum of three days in Siem Reap if you are planning on going to AnkorWat.
HCMC 4 days including Delta
HoiAn 3 days
Hue 3 days
NinhBinh ??
Hanoi 3days
Halong Bay 2/3 days
Siem Reap 3 days
Pnomh Penh 2 days
boat to HCMC
That takes care of your 3.5 weeks and only touching the highlights in two countries