Super Jolly

I can very strongly recommend an old Mazzer Super Joly.

Get a Mazzer Super Jolly with manual doser off eBay, buy a small hopper and cleaning supplies and a good little scale for weighting your shots. This fits your budget squarely. The Super Jolly easily produces a grind quality with stepless adjustments on the same level as a 2000,-EUR Mazzer and fits the ECM perfectly. The only inconvenience will be either manually dosing (weighting) or living with grinding a quarter doser and using 1 or two day old grinds. A Mazzer with electronic doser is out of budget (those usually go for 300+), have a weaker motor than the Super Jolly, grind slower and come with a PCB that just about now can go bad and is very expensive to replace (if you can find spares). The Super Jolly will last forever and once you are ready for the next upgrade you will easily sell it on for at least what you paid for it. I keep one of them on the counter just for French Press but have used it for espresso before getting another Mazzer with electronic doser. Congrats on the machine. This one is nice. 

When preparing coffee like French press or even drip filter machine coffee any 20 EUR bladed plastic grinder (chopper) can get you a reasonable cup of coffee. When preparing espresso though this all falls apart entirely. You need a very consistent, fine grind that is also prepared in a way that prevents the coffee beans from excess heat from the grinding action and ideally you want that grinder to last forever a veeery long time without needing spare parts, repairs and servicing. Consumer grade grinders that partly fulfill the requirements needed to produce the necessary quality grind start somewhere around 100-150 EUR (manual grinders) or electric grinders at about 150-250 EUR. These are consumer grade. Gastronomy grade (and luxurious consumer grade) grinders will typically start at about 450 EUR but more typically range in the 650 - 850 EUR price range. They are usually dimensioned significantly larger, using larger, slower rotating grinding disks or conical grinding mechanisms with slow rotating, high torque large motors, are made from premium materials with very durable construction (usually die cast housings over heavy duty metal chassis), high end models nowadays come with electronic precision dosers that precisely dose the ground coffee to fine adjustments of +/- 0.1g … These things are not trivial when aiming to produce consistently the best espresso based coffee possible. Unfortunately, this costs money. 

Very much no. Look for a used Mazzer mini or Super Jolly.

Super Jolly with single dose mod and SSP HU burrs

I have two grinders... A modded Mazzer Super Jolly for espresso and a non-esp Encore for drip. I think the ESP will handle everything you want to do.