PinkElephant

You are noted here as a Basic user. This means no access to support.
If this is wrong, there is a method to get this sorted out with support.
If not, no support.
About web clipper: There is no web clipper on mobile devices. There is the sharing function from the OS, that sends content into EN. What it picks is not defined by EN but by the OS. EN is responsible for the import.

You can do it from every client. Desktop: Help menu. Mobile: Settings, Support.
Or you follow this link:
https://help.evernote.com/hc/en-us/requests/new

You probably use an iPhone (you don’t tell …).
Go to settings, display & brightness or accessibility, and reduce the system text size.
The button to quit the screen will appear (in most cases).

No sync button on desktops - however it exists on the mobile apps. There is a thread with a vote to return it to the desktop clients as well.
Personally I am in favor of it as well - on rare occasions I had the v10 client on my Mac upsync, but not downsync for a while.
The note history is still there, accessible on the web and (if I remember well) on desktops. Works as before, a timeline of changes, and the option to reset the current note to an older version, or to create a duplicate note with an older version.

If something is missing from the share menu, sometimes restarting the device does the job.
Then it happens the target app is there, but hidden. You can check by scrolling the row of Apps ‚til the end, to the 3 dots icon. Click it, and a window will open where you can de-/select apps for sharing, and modify their sequence.

When it changes something you intentionally did, I regard it as a bug.
To bring it to the attention of EN and get it eventually fixed, the best way is to issue a support ticket.

Each Mac app that is installed through the AppStore runs in a sandboxed mode. It would take this thread too far, but in general sandboxing puts all interaction between an app, the OS and other apps under OS control.
This means an action by an app (like sharing) may not work, because it is not permitted by the OS. On the other hand this makes running apps more secure.
In iOS everything runs sandboxed. On the Mac, you can install via AppStore or via direct install, and for some apps both ways. In case of EN you install the very same code, but the AppStore restricts it on execution.
I would give the direct install a try.

Sure you can install and even run both clients in parallel. To avoid conflicting notes don’t work on the same note at the same time.
Install v10 first, and then legacy. Install the „grey“ legacy client, since it will never ask or try to update.
https://help.evernote.com/hc/en-us/articles/360052560314-Install-an-older-version-of-Evernote

The sequence is first change it, then save it
Open it, modify the parameters by clicking on the 3 dots, select filters, do the changes save it again - when you enter the same name, it will overwrite the existing saved search Done Edit: This works in the iOS client, not on desktop. 

The sequence is first change it, then save it
Open it, modify the parameters by clicking on the 3 dots, select filters, do the changes save it again - when you enter the same name, it will overwrite the existing saved search Done Edit: This works in the iOS client, not on desktop. 

For me it is no issue to run both clients. I just see myself moving more and more to use v10, because it simply has a better look & feel for me.
But in general, it should and one day must be possible to skip legacy altogether. Still way to go …

iOS (Mac is down for the Night)
settings support send feedback  Feedback will be reviewed, but usually not answered.

From what I see there is a limit of 20 personal templates per user. A sorting or grouping would be nice, but maybe together with a raise of this limit.

We had other threads here about similar problems.
There is a button hidden below what fits on the screen, if you have a rather large text font setting. To change this, go to iOS settings, display and brightness, and reduce the general text size on your iPhone.
You can then reach the button, confirm it once, and then the screen will not show again. You can reset you font settings to your liking again.

Currently no, at least not without a redesign. The reminder date „belongs“ to the note, and there is only one.
The workaround is currently to set it to the next issue on the list, and reset it as tasks mature.
+1 for the request

Stacks are a personal issue, to organize your personal view (!) of notebooks by putting them in one group. In fact stacks are not important, there are little actions that are stack-related. The organizing elements of EN are the notebook, the tag and the note.
Stacks are not copying to another persons account, even if the notebook is shared.
For example I share a stack with recipes with my wife, and she shares one with recipes with me. My notebook is about cooking, hers is about baking. We put new recipes in the respective notebooks by subject, not by owner. Once a new note is created in one of these notebooks, it reflects to all included in the shared access, the same with changes. If 2 people edit at the same time, you get a conflict note that one needs to review and sort out.
Both our recipe notebooks are in a "Hobby" stack in my account - my wife is not using stacks at all. No problem, but no way to share the stacks either. 
The concept of "spaces" is completely different. Spaces is the business account way to organize access to a group of notebooks. It has nothing to do with the stacks from personal accounts.

Not running EN v10 on Windows, but on the Mac there is a menu option in the view menu that toggles the note count on and off.

Here is how-to in the iOS (mobile) app, similar on Android.
Hit the search icon In the search window, hit the filter icon top right Now you can directly select tags, notebooks, dates, content and more from the popup menu.  When you select a notebook („is located in“), the search is limited to this notebook.
Maybe the desktop clients do it a little differently, had no time to try.
When you already start from a notebook view, it is even more easy: When you hit the search icon and the search screen opens, the notebook will show in a preselection below the search field. Just tap on it (it turns to solid), and the search is restricted.

 

Most likely the legacy client has collected some residue in its local data base.
When you open the web client, it should match the v10 clients count.
Go to the legacy client. Hold down the opt key and click on the help menu. There now are additional options at the bottom of the menu list. They serve to optimize the local data base. Run them to make legacy check and correct database errors.
Another issue: Do you have received shares from others ? Maybe you need to reactivate them in v10 to make them count.

Bug reports here do not reach their recipient - this forum is mainly user to user.
Best way to get this fixed is a support ticket. If you reproduce the problem on your Mac, then save the activity log (EN help menu) and finally issue the support ticket, you can send the log file. This probably helps in a case like this.

Go to the help menu, option keyboard shortcuts, or click on the little keyboard symbol down at the side panel.
The shortcuts list will open. You can modify the first shortcuts in that list, the general ones.

Search has not really changed from the legacy client. Both send the search request to the EN help pages. There selection could be better, as @Mike P has already pointed out.
The only option that is not implemented in V10 is the assistant that shows where to find a function in the Macs menus. In legacy you can choose it after entering the search string, and it will open the menu and show the function with a blue arrow. This is pretty standard to Mac apps - whether it was not yet implemented, or does not work with Electron apps (the framework in which EN is nested) I does not know.

You show here as Premium.
When downgrading, the best way is to unsync devices BEFORE the downgrade. Probably a clear cut does it all: Quit all devices, using the web client. Quit the web clipper on all browsers. Quit access for 3rd party apps as well, that may have access through the API. Then log out of the web client as well.
Wait maybe a day, to let everything trickle down the servers. Then only log into those 2 devices you will be using when on Basic. When Basic is active, you should be able to do things within the limits of the plan. It just goes beyond everything when you enter Basic with the Premium setup still in place.
Here is more about devices and the Basic plan:
https://help.evernote.com/hc/en-us/articles/218558068 Background: The limit of unsyncs per month was AFAIK introduced after "clever" users did client hopping to get around the Basic restriction. So say "thank you" to this minority group of users who stretch the limits until everybody feels the pain.

The entry into the start menu is not automatic. The installer needs to do it as an extra step, which can be deactivated as well.
Maybe it was missing from the installer, or you disabled it accidentally when removing one of the many checkmarks that appear during an install.

Set the automatic lockscreen to never, open the EN app and keep it in the foreground. If necessary plug it into the charger. Let the download run. The download of 19GB to my iPhone took appr. half a day - maybe let it run at night.
It will only load when the app is in use, or at least open and active. Usually the loading of new stuff is pretty fast.