Yes. To be assessed means that if you can't negotiate out of court, they can prepare a formal Bill of Costs and prepare for a detailed assessment hearing - where a judge will assess costs. So for now, negotiate. You do not have to pay them exactly what they're claiming.
Look at the rates they're charging. Grade A, B, C or D. Check the court hourly rate website - Google. It'll tell you what hourly rate each grade of solicitor is entitled to charge depending on the court and / or the Defendant's location.
You can find out what grade someone is by looking at the 'find a solicitor' website. Grade A = 8 years post qualification experience, Grade B = 4 years PQE etc etc.
You get to negotiate and you can make offers pursuant to Part 36. Evaluate the work they claim to have done and how long it ought to have taken them. It's pretty hard given you're a LiP but it's how the court will deal with it.
If they're in breach of the order you could apply at your own cost and risk. If what they've served is not what you're supposed to have received then you could make an application but make sure you're right otherwise you're looking at paying their costs of the application as well.
I don't think there is a format for a schedule of costs but I could be wrong.
Essentially, there is a requirement to serve a bill of costs. A form N260 is normally used in advance of a hearing where summary assessment is sought at the hearing. If no proper statement of costs is filed in advance then detailed assessment is ordered. In that case, a bill of costs is required and it must include the matters set out in the relevant paragraphs of Part 47.
I concur with the advice to make an offer on their costs if possible. A general rule of thumb is that a person can expect to recover about 60% of their costs on a standard assessment basis. However in reality it will be necessary to look at factors such as the guideline hourly rates and whether the amount of time spent on any particular task was reasonable.
I don't think you should ignore it. It would be reasonable for you to write to them and tell them you don't think they've complied with the order but bear in mind, you'll pay for them to have that formal bill prepared.
The form you've referred to is a 'model'. Doesn't have to be followed but it usually is.
You could put them on notice of an application but I don't know what your prospects in an application would be. Under Part 47.7 of the CPR, a party is allowed months to commence detailed assessment from the date of judgment. If I was defending an application, I'd argue 7 days for a full bill isn't reasonable and that the CPR should be followed.
Basically, I think you should start to negotiate rather than do nothing and incur further costs.
A Bill of Costs usually comes with a Notice of Commencement (of assessment proceedings) and you have (I think) 21 days to file 'points of dispute'. If you fail to file POD, they can apply for a default costs certificate.