Zhakharov:
The reason my population is usually low when I switch to Republic is because I have been pumping out settlers and workers during the expansion phase. You should still switch to Republic even if you don't yet have the pop to support troops adequately. Any cities over size 7 will probably starve back down to size 6 or less during anarchy, so there is no 'good' time to switch. There will be a recession period whenever you make the change.
I find this tactic generally ill-advised. If you switch to Republic at such an early time, an unfortunate attack by the Mayans, Iroqouis, or Romans could finish your game then and there. Your Republic will be reeling from the WW issue and your unit support is so low you won't be able to muster a counter-offensive.
It isn't true, either, than cities over size 7 will starve back to size 6 during the anrachy period, and I find this especially so because I make provisions for it. Cities starving at all during a government shift is unusual for me.
If you have good infrastructure and good planning, I find that you can almost always hit the ground running as soon as you shift into Republic, whether you're shifting from Monarchy or Despotism.
I always find in my games that the earlier I switch to Republic, the faster I can accelerate away from the AI in production and research.
Always with the proviso that no one's going to attack you before your cities hit size 7
and you can churn out the military. Going into Republic with a minsicule military is like sending unescorted Settlers out to found undefended cities. It's phenomenal if it works, but spectacularly game-losing when it doesn't.
Defensive units in core cities under a Republic are a waste of time. It is much better to have extra offensive units patrolling the borders instead. Like I said earlier, the only core city that needs defending is the capital.
With standard map sizes, an enemy civ overseas with a couple Mounted units can raze your core into the ground with hardly any effort. The AI actually does this on occasion. The maddening thing is that they will occasionally pull such an underhanded stunt while on RoP agreements with you. When this happens, you will bitterly regret not having units in those cities. Unless, of course, you happen to see this as an occasion to simply happily reload the game.
I can't stress it enough. Defensive units in all cities are a must in any government form.
3 to 4 units per city is a lot of troops when you also have 3+ workers per city.
3+ Workers for every city is probably overkill, especially when many of your earlier tiles were worked by early workers. 2-3 Workers per city generally works for me until I have to make about 4-5 per during the early Industrial era for Rail. Even with 3+ workers per city, 3 to 4 units per city is not at all "a lot of troops". I usually station at least 2 units per city as defensive garrison, and as a defensive unit buffer for when my production doesn't match unexpectedly swift army advances. I use 3 on Monarchy for much the same reasons - defensive unit buffer.
Aside from this, I'll usually have 1 mounted unit for every 2 cities, and possibly 1 SOD worth of components in defensive stances throughout the region for every 4 strong productive cities This includes Swordsmen, Horsemen, Catapults, and extra Spearmen. Minimum SOD is 3 Swordsmen, 2 Catapults, 2 Horsemen, 2 Spearmen. That's 3-4 units per city on the average, minimum, ideally speaking. If I'm thinking about war, I'll expand this to include 1 SOD for every major city I'm planning on taking over, serving to as much as double the unit count to as much 8 units per city. Needless to say, this costs prohibitively, even on Monarchy, and absolutely untenable on small Republics. This force, of course, is meant to wage war, not stand around, so they're projected to eventually pay for themselves in terms of additional cities, luxuries, what have you.
This is especially true when I'm planning invasion as a Republic, since I can't afford losses, and I can't afford long wars. If I can't see my way to my war objective within 20 turns of declaring war, I will not start one under Republic. The sooner the objective is met, the better. A 1 turn war in the thick of the Middle Ages is possible. Destroy the civ on the turn you declare war. Five turn successful wars are extremely doable with 8 units per city on the cusp of declaration of war.
So no, I don't consider 3-4 units plus 3+ Workers per city "a lot of troops". On Monarch and even on Regent, you'll find that you will have to take substantive losses in taking cities, whether in units, or in War Weariness. It's not unusual for a standard SOD of 3 Swordsmen, 2 Horsemen, and 2 Catapults to sustain heavy losses on the assault, especially on well-fortified cities boasting as many as 6 to 8 well entrenched defenders. It's entirely possible that they will fail to meet their objective against a city on favorable terrain, with good unit defense.