Couldn't find this info nor a thread for this, creating one. I and others I know wish to know if items left to be claimed will be lost and when, or how that will all work.

I'm not sure if you mean reward items you've unlocked but forgot to claim OR reward items that you haven't unlocked yet because you've not moved far along enough to get them (e.g. starting the season late). In either case, I suspect you lose the chance to claim them from the reward track because once the current GS season ends, all rewards will become available to buy for all players on the vendor Ki'at until the next Season starts:

Once the Season concludes, Ki’at’s inventory will open to everyone, giving Subscribers the opportunity to purchase rewards they might have missed out on. Ki’at Thavo’s inventory is Seasonal, and will completely refresh with all new rewards at the start of Season 2.

This is from: https://www.swtor.com/info/news/article/20210426-1

The things like all the random gear and gifts that i didn't want cluttering my inventory as I leveled GS, what happens to them? I only claimed CC and GS tokens, mostly.

Claiming everything you've unlocked before the season ends is the best approach, because I suspect that once a season ends anything you've unlocked but not claimed will be gone and then you won't be able to claim them! Make sure to use all of Altuur's companion gifts on a single character so that you get Altuur to lvl 50 on one character. That way you unlock the option to purchase the companion gifts for credits on your other characters.

The things like all the random gear and gifts that i didn't want cluttering my inventory as I leveled GS, what happens to them? I only claimed CC and GS tokens, mostly.

You won't be able to claim them once the season ends, but they may become available to buy on the vendor. Claiming everything before the season ends is probably the best option. Armour and weapons open in collections on the character you claim them on and you can unlock the two armour sets and all the weapons account wide for 10 CC each. Decorations go into their own tab once used so none of these should clog up your inventory.

Make sure to use all of Altuur's companion gifts on a single character so that you get Altuur to lvl 50 on one character. That way you unlock the option to purchase the companion gifts for credits on your other characters.

I found out by accident that these gifts unlock to buy on the vendor with credits if you level Altuur to 50 using an insta-level-50 influence token. I made a couple of level 70 boosted alts on a different SS & SF (my main is on DM) so I could do GS there. I wanted level 50 companions on my new 70s for crafting so claimed Altuur from collections on them and used a Commander's Compendium (the one you can buy from the Conquest Vendor) on him. After that I could buy the gifts from the vendor for credits (rather than tokens).