Does the game allow the player to send in 2 or more lances? I didn't in the storyline, and before I start a career I was wondering.

No, nut unless you MOD the game. I would love to be able to call for reinforcements like the enemy does.

there are some missions that have two or three lances on your side but you always only have control of your [upto 4] lance and the others will do some dum shit that you have to save them from

I added the mods Bigger Drops and Mission Control, to enable the option of dropping a second lance. In mid-game, it does make the fights quite a bit easier if you drop an entire lance, so I rarely use it unless I want to have an extra mech for scouting. I did have to make a few edits to one of the .json files in one of them to get it working correctly, and I had to download them from their GitHub page.

Some of the larger mods like RogueTech also have the option for a second lance already preconfigured.

Mechcommander had drop weight lmiitations. Would be nice to implement such a system, maybe you want to start with light and medium mechs and call the heavy armour when in desperation.

Few missions in BT, maybe 2%, have weigh limitation, so it is advised to keep at least one mech from all weights to fill the best these rare missions.
Edit: Maybe these missions come from one of the 3 DLCs.

At least one is from the original, I remember because the steel claw mech was so broken for so long.