I've done a small amount of mixed climbing but I'm not an expert. Still, it's possible you will find something in the following to lighten your load.
You stated that you usually go solo so either you're doing roped solo or this question refers to partnered climbing.
Unroped soloing is definitely your first choice for going both light and fast.
Start with your heaviest item and that which you carry many of. This means rope and biners.
ROPE - I like my 7lb 13.5oz 60M 9.1mm Beal Joker. Since it's rated for single, double and twin use I can run it out on most terrain and for short difficult sections double it and use double or twin technique for ~30M pitches.
If you want to go lighter and you're highly skilled at knowing when to hang it out and when to be cautious, use the Tendon 7.8mm twin/double rated rope (~5lb7oz @ 60M). If you posess the appropriate skill/risk profile for this application, use it as a single rope on the easy sections and double it for short, difficult sections, high chance of rock fall, sharp rock, wandering leads or frequent ledges where you want low rope stretch. Obviously you need to have precise pick and crampon placements to ice climb with a single rope (or else you will soon find yourself soloing).
The lightest way I know to make full length retrievable rappels is to tie a rappel ring into the end of the rope and pass the other end of the rope through the anchor and the ring and tie a full length 3mm accessory cord to the ring. Rappel the single rope and then pull on the accessory cord to retrieve the rope. With the Tendon Master, 3mm cord and rappel ring your total is 6lbs for 60M rappels.
BINERS - Upgrading to Wild Country Helium biners with the clean wire feature saved me just over 1lb for the 40 on my rack. you can save another 5oz if you're willing to go with a smaller biner and lose the hooded nose feature.
ROCK PRO - Ray Jardine's "Friends" reinvented parallel and flaring crack climbing. Because these tools are so cool, climbers ditched their hexes and started dragging heavy SLCDs everywhere. I find that in the backcountry I can almost always find a placement for a slung feature, a stopper or a Tricam (lighter and more versatile than hexes). Unless you're doing a route that truly requires SLCDs, leave them at home.
RUNNERS - I haven't carried a closed sling in 15 years. Once I tied my first rabbit runner (single strand webbing with a loop at each end) I never went back. They have twice the versatility of loops at the same weight. Fortunately Mountain Tools now makes "snake runners" out of nylon/spectra @1.4oz. I also have double length ones made in a contrasting color. They offer a stiffer ice snake runner too for threading ice anchors.
I find a 5.5mm spectra cordlette at each anchor to make for light, fast, strong, equalized rigging.
ICE TOOLS - If you're not spending much time on rock, C.A.M.P. makes an all aluminum 14oz crampon and 9oz, 60cm axe. If you're ready to hang it out like Dr.J and/or modify your technique to limit load on your equipment, try the uncertified ULA-Equipment Helix CF/aluminum 5oz axe.
HELMET - 8.8oz Every. Black Diamond just came out with a model that is .5oz lighter but lacks a headlamp attachment system.
ASCENDERS - Are you using the lightest model available? There's a tremendous variation in weight between manufacturers and models. Can you get away with the prussik knot? It's a model of economy, versatility and light weight.
GRI-GRI - This is a real clunker. At 7.9oz there's gotta be a lighter way to achieve whatever you're doing with this boat anchor. A muenter hitch weighs 0.0grams.
PACK - Once you've got your weight and volume down you can move into a smaller, lighter pack. And while you're at it, stay away from those armored, lifetime use packs. If I can't tear it up with a few seasons of hard use, it's too heavy. My 1lb 3oz Go-Lite Gust is sturdy, has adequate features and sufficient volume for way more than I want to carry.