Three Circuits In One Box
You can have multiple circuits in one box. But you have to:

Keep all matching hots, switched hots and neutrals of any circuit together
Keep all neutrals of different circuits separate (nobody would ever think, I hope, of combining hots of different circuits - and neutrals should not be any different (except MWBC...))
Connect all grounds together

So it sounds like the existing setup is a bit strange but not necessarily a problem, in and of itself. Grounding may be an issue, and is the simplest part, so let's tackle that first:
Grounding
Grounding can be through a ground wire or through a metal connection - e.g., metal conduit. In your case, you said &quot;all 6 ground wires were capped under one nut&quot;. That sounds like all 3 cables in and all 3 cables out. You're set for grounding.
Box Fill
A box can only hold a certain number of wires and devices. The exact number depends on the dimensions of the box, the size of the wires (12 vs. 14) and a number of other factors. It gets complicated. Assuming that the existing setup is code-compliant, it is quite possible that replacing a simple switch with a timer may cause either code (calculated values) or practical issues with box fill. If that is the case (or &quot;just to play it safe if you're not sure&quot;) removing one of the switches would likely solve the problem - giving you more room to work. However, removing a switch leads to...
Switch Requirements
Most rooms are required to have a switched light. Or rather, they have to be designed for a switched light, though nobody can force a homeowner to put in a light (or to change a light bulb that burns out). This can be done using a switched ceiling fixture or (as was the case with my house - first thing I got &quot;fixed&quot; when I moved in) a switched receptacle for a plug-in lamp. That is why you have a switched receptacle.
If you have another switched light in the same room as the switched receptacle then you should be fine. If you do not have a switched light then code may actually require you to keep that switched receptacle (or, alternately, change it to a switched ceiling fixture).
If you do remove a switch, best practice is to either remove the wires or cap them with wire nuts (so they could be used again in the future, and disconnect the wires at the breaker or prior junction so that you don't have extra, unknown, live wires in the box.
Combining Circuits
You may be able to combine circuits. That can help with box fill (both code and practical) because then you can remove some of the cables. However, you need to make sure that you are not going to end up with overloaded circuits. That is generally not a problem with LED lighting, but it is still a consideration. When you have any 3-way circuits you need to make sure that you are combining only at the &quot;incoming power&quot; switch and not at the &quot;other switch&quot; as you could end up with some wacky (and possibly dangerous) situations if you mess that up.
Neutral Problem
Harper pointed out that your neutrals are currently combined between circuits. You can't do that! You will have to match up each neutral with its matching hot/switched hot. Then combine each of those pairs of neutrals together. In the end, the number of neutral groups should be the same as the number of circuit breakers that are related to circuits in this box.
Timers and Neutrals
Many timers, smart switches and other devices require neutral. Code now requires neutral in most switch boxes for that reason. Based on your description, that should not be a problem here, with the possible exception of the 3-way switch (again, if it has &quot;incoming power&quot; then it is fine but the other 3-way likely does not have neutral).