For pike rods I've come to prefer a baitcaster mostly because I run heavier line and lures. I've made a couple of my open water bass reels work for ice that's been easy. I'm actually in the process of assembling a couple more combos so I've been shopping too. My plan is eBay for a couple of cheap BCs. First, you're not casting so you don't need 11 million ball bearings or crazy high retrieve ratio. Because there's a freespool button even the drag doesn't need to be great, just serviceable. Next, that reel will be little more than a line holder so consider capacity. 50 of 30# Dacron is usually plenty for a tipup so you can spool with that if you want or pile on some superline or mono. It won't have to be a huge reel, no bigger than a "200" series.
Just missed a used Quantum EX501 (lefty) for under $15. Someone nodded off waiting to snipe it
. There are some Bill Dance reels that look like they can be had for around $20 new that are pulling me unless something else pops up. If you're planning on $40-70 new you'll get something decent. Stick with a brand you know if you can: Daiwa, Shimano, ABU, even Bass Pro brand aren't bad entry level reels.
Walleye reel? Pflueger President runs $50-60 and are great reels