Reddit mentions of Washington Scrambles: Best Nontechnical Ascents

The Mountaineers, like you already mentioned, may be your best bet. Washington Alpine Club is another option, though they are also passed the registration period for classes this year. You could certainly continue climbing and gaining experience on things you feel comfortable on, lots of snow climbing around here doesn't require glacier travel or even much experience, just use of ice axe/crampons and a love for long walks. Examples include Mt. Adams, Mt. St. Helens in early season, and basically anything in the Washington Scrambles book (highly recommended) during early season.

You could also try posting over at Cascade Climbers for partners - the only thing I recommend is being very upfront about your level of experience (or lack thereof). If you overstate your experience, trust me, your partner will know, and they'll likely decline to climb with you again.

My recommendation is that you should get your own systems dialed before asking for partners/team. You mention you have steep snow (and ice?) experience? I'm going to assume you don't have much glacier travel or crevasse rescue experience, and likely no alpine rock experience either. Lacking these skills, what you are asking for is essentially a free guide service, or someone to teach you all the technical stuff while assuming all the risk.

That said, my recommendation is to actually hire a guide for a 'skill session.' You can often find one, two, or three day crevasse rescue courses, and then supplement that hands-on knowledge with some old fashioned book reading, as well as tons of practice on your own time. Once you get everything dialed, then post on Cascade Climbers, and I bet you'll have good luck with someone willing to take you out.

May is still fairly early season, the vast majority of ascents happen in July and August. Since it's early season and PNW weather is what it is, you're much more likely to have bad weather than if you waited a few months. Depending on how long you'll be here you may get a great weather window or could have a week+ long period of low visibility and rain.

Would you trust your friend to rescue you from a crevasse without any assistance? If not, they're probably not the best partner to take on this climb. While many of the crevasses will still be snow covered in May, they may be starting to melt out and thin snow bridges won't be as apparent. Mt Rainier is not a 'technical' climb nor is it especially high, but it certainly is a test of fitness and has its fair share of objective hazards.

Depending on conditions, any of those routes could be the right one for you. All are relatively easy as far as glacier climbs go.

I'm not going to go into depth on the other questions - you can do your own research there. RMI and IMG are local guiding companies that can provide resources and make recommendations.

Washington has a huge selection of hikes, from PCT section hikes through the Cascades to long loops in the Olympics. WTA is a great resource for finding the right one for your trip. I'd recommend Washington Scrambles as a good book for non-technical summits in the area.

https://www.amazon.com/Washington-Scrambles-Best-Nontechnical-Ascents/dp/1594858403

Kaleetan Peak is a good one. Lots of good stuff in the Enchantments.