I just did a quick mental inventory; thanks to my own interests and a bunch of years as a Cub Scout and Boy Scout adult leader, I have:

An ancient Eureka 2-person Timberline - A-frame style, comfortable for 1 adult or 1 friendly couple or 2 younger scouts. Sort of self-supporting, more complicated to set up in rain/dark. Heavy for the size. I let my son use this one.

A Eureka 2/3 person (?) pop up (Hexagon series?). Self supporting and cheap. Best you can say about it. My kids have use of this one, too.

A Sierra Designs Clip Flashlight CD (the 1 person version). Used it on a number of Scout overnights and a 4 night C&O Canal trip. Easy to set up, very comfortable for me at least in cold and wet. Cozy; you need to be a bit of a contortionist to get in/out and dress/undress inside. Sufficient inside storage for 1 person's gear unless you are tall. Very lightweight for the size. Was able to fold small enough so only the pole bag had to ride outside my panniers. Bought mine on sale from REI, Campmor and others may have older year models on close out. This is my favorite if size/weight are primary and I don't need much interior space.

REI Half-Dome. Comfortable easy to erect 2 person tent. Self supporting. Weight not bad for 2 people, a bit heavy for one person vs. the Sierra. 2 doors with awnings for gear storage. Try to get one on sale or end of year close out. Rentable if you want to try one from local REI stores. My favorite if I want space, comfort, and not so concerned about the packed size and weight.

Tarp - used one ONCE on an early December camping trip (temps dropped into mid teens). Yes, ALL the Scouts and other adults thought I was NUTS. If it had been windy or snowed, they would have been RIGHT! (and I would have pulled a real tent out of my car). To use with a bike instead of hiking poles, I'd use the bike as a support (guy it with rope and stakes) and rig the tarp lean-to style or with a nearby tree for the A-frame look. You still need a ground cloth and there is ZERO bug protection. With the mosquitos and deer tick problems in the Northeast and Midatlantic, I wouldn't use one from mid April (maybe earlier) until the first hard frosts. Might be OK with a bivy sack for wind/precip and bug protection.

Comments on the Eureka Solitaire and Gossmar - I had considered them before getting the Clip Flashlight. One of the scouts had a Solitaire on the C&O trip. He thought it was OK, but I observed: Not significantly easier to rig than my Sierra Designs. Basically ZERO room for gear inside. Specs show no huge weight advantage vs Sierra Designs. Personal: too cramped for my taste. Significantly cheaper vs Sierra Designs but an end of year model closeout sale and my REI dividend helped.

If there's a backpacking store you can visit, check their tent displays and spend some time wriggling in and out of them. Some stores (REI, EMS, Hudson Trail) have very knowledgable staff who can help alot with advice and suggestions. Good luck.