Specific answers to your questions:
Does suspension reduce acceleration?
Yes, for two reasons. Suspension causes bob, where some of your pedal power goes into flexing the suspension with each pedal stroke. That power does not spring back out and help you, it is lost. This can be helped if your suspension has a lockout function, which is like an off-switch.
The other loss is that suspension adds weight to the bike, so you're moving a couple kilos of metal and oil all the time. Less important on the flat, but still there.
Does it reduce max speed?
Yes–if you can't lock out the suspension, you'll be losing some power. If you lock out the suspension, you're still carrying it with you on the bike.
But no, smaller amounts of added weight generally don't have a lot of effect on the top speed, all other things consistent.
Does it reduce braking power?
Yes. The added mass means you have more bike to stop, and physics says you'll take longer to stop an increased mass.
Also front suspension loads up when braking, helping to move the bike's weight forward and making the back wheel brake even less useful.
Additionally, suspension loaded under braking can have some unexpected side effects if you're not expecting them. Loaded brakes will unload as soon as the bike's momentum is stopped, and it will push your bars upwards. If you're not steering straight, then things get all additive-vectory and you get pushed to the outside of your turn. This is fine if you're expecting it, but potentially dangerous if you forget or didn't anticipate.
For a paper route or similar, you'll be going slow and stopping at most mailboxes. You want a bike with
- Flat pedals
- Relatively low seat (lower than optimal for road) because you'll be putting a foot down more often
- Flat bars - no need for aero drops here.
- Rigid front fork - makes installing a front parcel rack easier
- Rigid frame - Same for a rear rack - those cantilevered seatpost ones don't hold a lot of mass.
- Reasonably wide tyres - 2" will be ample for most of your needs.
- Puncture-resistant tyres (or get them on your first tyre replacement) because punctures on a loaded delivery bike would mean a complete un-load which is no fun!
- Cheap - don't get an expensive bike for a paper run. If you're going longer and further with your deliveries then reconsider this.
ANSWER You want either a BMX or a 90's style rigid MTB, with strong front and rear racks, and mudguards to keep it cleaner. (but not the knobbly tyres in the photo)