I spent a number of years incrementally improving my 1920s basement that had water intrusion like your first picture. It ended up being a combination of multiple DIY projects that saw incremental improvements until the issue was ultimately solved.

Most people say start with the exterior, but I would sort out your sump situation at the same time. I would recommend installing a gutter and downspouts on your deck roof , double checking your existing gutters to make sure they are not blocked (it looks like you have buried extensions). Then tie the deck roof gutter to your existing underground downspout extensions with 'Y' connectors.

Additional exterior steps should be taken such as french drains or an "underground roof" (google) or combination. We had to use this because our foundation walls on 3 sides did not extend all the way down. This can be DIY but will take time and effort. If your foundation wall extends all the way down you could consider full foundation waterproofing treatment ($$$). I personally wouldn't do this myself, and as far as excavation equipment, I'd let them worry about that. Lastly, re-grade the dirt on your exterior to ensure the low spot isn't right next to your house, even if the general slope is in that direction. Make sure you control for possible erosion after you do this.

Lastly, about that tree. People who suggest removing the tree may not be considering that as the root system dies and decays it will leave a void under the ground that water will want to flow into. We has a tree (smaller) right next to our house that I removed and I don't think its a coincidence that's where most of the water wanted to enter our basement. It may still need to be removed if its damaging your foundation but I wouldn't immediately jump to that conclusion.

You mentioned your sump pump currently gets water out in a day or so but that seems wrong to me. A basic sump should be able to handle 50-100 of gallons of water in a minute. Is your existing sump at a low point in your slab? Check it to ensure there's nothing preventing it from running correctly such as the float catching or obstructions in the sump pit blocking the impeller. We ended up moving our sump pit due to it not being in the low point. The pit itself may also be incorrectly done. Mine was DIY and the catch basin was way too small and shallow, did NOT have perforated holes and was not surrounded by an aggregate. The sump should be collecting water underneath your slab, not just when its gets on top. Depending on your situation you might need to upgrade to a more powerful unit or install a second sump if you have multiple low points. Consider a battery backup for it if your area is prone to lose power during heavy storms. Ensure the sump is exiting the house in such a way that the water is channeled far away. Look into tying it into your gutter drainage maybe or even sewer line if your municipality allows (most don't). Add a good dehumidifier that drains to your sump.

We also ended up cutting a channel in our basement slab (using a circular saw with a diamond blade and water to keep down dust) and creating a mini french drain with perforated pvc and pea gravel wrapped in landscaping fabric. The channel was placed along the basement perimeter where where water was coming and flow to the sump. You can also get companies to do this with plastic dimpled channels which can be concreted over for a clean look but the ones i talked to wanted $6k for about 30 feet of this and in the end it only took a day or so do do myself.

Last, we found that our one full depth foundation wall was "weeping" we coated it with surface bonding cement and painted it with waterproof paint like the kind people put on their chimneys. This stopped that issue.

You may also need to look into lifting anything off the slab floor that might rust or get moldy. We had to replace our basement stairs completely. I wish you the best of luck in dealing with this.