Once a baby maple seedling is 'topped' it is dead. I've never seen 'suckering' from maples of an old coppiced maple or newly cut down maple tree. Those suckers arise pretty much right below the soil line of the stump. Not out in the lawn, I've never ever had problems with germinating baby trees. As for suckers that would be Aspens...those do sucker but only if there is a drought in progress. Keep them watered and they won't sucker.
Baby grass is usually not fertilized until AFTER the first mow. Do not worry that the baby maples are sucking up chemistry out of the soil because those maple babies have a bunch of food provided in that samara until they too are able to make their own food. THEN all plants needs a little balanced fertilizer added. Lawns are usually fertilized 4X a season. Lots of growth and regrowth.
I am more worried about the height you will be mowing. Lowering your deck so that you can chop up baby maples is not at all a great start for your blue grasses. Raise your deck to cut no shorter than 3". 3 1/2" is best. Do not leave clippings on the lawn, bag them and compost.
Do not water until you see your footprints stay down in the grass. Walk on your grass and if you see your footprints stay down, water VERY deeply 4 - 6 " deep (use a shovel to check the soil profile and note the amount of time, you could also use kitty cat cans all over your lawn and water for 15 minutes. Check the depth of water in the soil and check the level of water in the can. If it is 1/4 inch that means you've used 25% of your 1" allotment of water per week. Clay soil holds onto the water better than sand. The rule of thumb is 1" of water per week. You want to train your grass to grow deep deep roots by allowing the soil on top to dry downwards causing the roots to grow towards the moisture. Shallow watering is fine for baby grasses but after that first mow it is time to start training your grass crop to have deep large root systems.
In order to generate and grow and have healthy root systems those grasses need top growth. The photosynthetic green part of the plant has to be no shorter than 3 to 3 1/2 inches to keep your blue grasses vigorous, dark green, drought tolerant (when your neighbor's lawns turn brown yours will be dark green and moist and you'll be using less water than your neighbors, honest). That top growth if cut too short will not be able to make enough food for itself, they have genetically huge root systems and without 3 3 1/2 inches of top growth the grass is stressed. Weeds are able to germinate without the shading of 3 to 3 1/2 inches of grass, they are getting lots of sun when you cut at 2 inches. You are literally 'scalping' your grass at that height. No matter how much fertilizer you throw at your lawn, if it is cut too short (we are discussing cool season grasses) there are not enough photosynthetic FACTORIES to make food. The weeds thrive on the extra fertilizer. If you throw 'moss killer' on your lawn you are lowering the pH. Lawns need a more alkaline pH. Moss is just an opportunistic plant that when it finds readily available and consistent watering (shallow everyday watering) it will fill big holes in the lawn and cover bare soil. Moss is a great 'indicator' of poor maintenance practices.
I would completely ignore those baby trees. Never in all my years of maintaining thousands of grass crop acres have I ever had a problem with baby trees.
Have you fertilized your lawn? How much did you use? What formulation? What product? Let me know. When you are out and about shopping be on the look out for Dr. Earth's Lawn Fertilizer. There are others similarly formulated. Extended release. Greens slower (healthier) is balanced for the time of season into 2 or 3 formulas (low nitrogen for the fall)...costs more but pays off because you fertilize less; maybe 2 or 3 times a season not 4. It also has bacterias and fungi you want in your soil. Thatch eating bacteria?
Bag clippings. Mow no shorter EVER than 3 to 3 1/2 inches. Do not use any chemicals such as moss killer, lime, weed and feed, or turf 'builder'. Without talking to us first, okay? You do more harm than good every time you use a 'product' advertised to make great lawns. Aerate once per year. Easy to rent and do leave the plugs right where they lie.
Water only when you see the blades of grass you stepped upon stay down. If you follow these maintenance practices that successful landscape maintenance companies follow you will OWN your lawn. Think about maintaining rich people's lawns and landscapes. You do not mess around with 'patching' up problems. You simply don't make problems happen. Pesticide in no Universe I understand is ever necessary. And I have been a commercial pesticide applicator for over 3 decades.
How long did you plant your lawn seed? Was it pure blue grass? Did you ROLL your lawn bed? What did you use for fertilizer and how much?